 16.16.1: A vector field on is defined by . Describe by sketching some of the...
 16.16.2: Sketch the vector field on given by Fx, y, zz k 3
 16.16.3: Imagine a fluid flowing steadily along a pipe and let be the veloci...
 16.16.4: Find the gradient vector field of . Plot the gradient vector field ...
 16.16.5: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.6: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.7: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.8: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.9: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.10: Sketch the vector field by drawing a diagram like Figure 5 or Figure 9
 16.16.11: Sketch the vector field by drawing a diagram like Figure 5 or Figur...
 16.16.12: Sketch the vector field by drawing a diagram like Figure 5 or Figur...
 16.16.13: Sketch the vector field by drawing a diagram like Figure 5 or Figur...
 16.16.14: Sketch the vector field by drawing a diagram like Figure 5 or Figur...
 16.16.15: Match the vector fields with the plots labeled IIV. Give reasons fo...
 16.16.16: Match the vector fields with the plots labeled IIV. Give reasons fo...
 16.16.17: Match the vector fields with the plots labeled IIV. Give reasons fo...
 16.16.18: Match the vector fields with the plots labeled IIV. Give reasons fo...
 16.16.19: Match the vector fields on with the plots labeled IIV. Give reasons...
 16.16.20: Match the vector fields on with the plots labeled IIV. Give reasons...
 16.16.21: Match the vector fields on with the plots labeled IIV. Give reasons...
 16.16.22: Match the vector fields on with the plots labeled IIV. Give reasons...
 16.16.23: If you have a CAS that plots vector fields (the command is fieldplo...
 16.16.24: Let , where and . Use a CAS to plot this vector field in various do...
 16.16.25: Find the gradient vector field of f fx, yxe f Fx0
 16.16.26: Find the gradient vector field of f fx, ytan3x 4yxy f
 16.16.27: Find the gradient vector field of f fx, y, zsx f x, y, zx 2 z 2 23....
 16.16.28: Find the gradient vector field of f f x, y, zx cosyz2 y
 16.16.29: Find the gradient vector field of and sketch it fx, yx 2 y f
 16.16.30: Find the gradient vector field of and sketch it fx, ysx 2 y2 fx
 16.16.31: Plot the gradient vector field of together with a contour map of . ...
 16.16.32: Plot the gradient vector field of together with a contour map of . ...
 16.16.33: Match the functions with the plots of their gradient vector fields ...
 16.16.34: Match the functions with the plots of their gradient vector fields ...
 16.16.35: Match the functions with the plots of their gradient vector fields ...
 16.16.36: Match the functions with the plots of their gradient vector fields ...
 16.16.37: A particle moves in a velocity field . If it is at position at time...
 16.16.38: At time , a particle is located at position . If it moves in a velo...
 16.16.39: The flow lines (or streamlines) of a vector field are the paths fol...
 16.16.40: a) Sketch the vector field and then sketch some flow lines. What sh...
 16.16.41: Evaluate , where consists of the arc of the parabola from to follow...
 16.16.42: A wire takes the shape of the semicircle , , and is thicker near it...
 16.16.43: Evaluate , where (a) is the line segment from to and (b) is the arc...
 16.16.44: Evaluate , where is the circular helix given by the equations , , ,...
 16.16.45: Evaluate , where consists of the line segment from to , followed by...
 16.16.46: Find the work done by the force field in moving a particle along th...
 16.16.47: Evaluate , where and is the twisted cubic given by
 16.16.48: Evaluate the line integral, where is the given curve
 16.16.49: Evaluate the line integral, where is the given curve
 16.16.50: Evaluate the line integral, where is the given curve
 16.16.51: Evaluate the line integral, where is the given curve
 16.16.52: Evaluate the line integral, where is the given curve
 16.16.53: Evaluate the line integral, where is the given curve
 16.16.54: Evaluate the line integral, where is the given curve
 16.16.55: Evaluate the line integral, where is the given curve
 16.16.56: Evaluate the line integral, where is the given curve
 16.16.57: Evaluate the line integral, where is the given curve
 16.16.58: Evaluate the line integral, where is the given curve
 16.16.59: Evaluate the line integral, where is the given curve
 16.16.60: Evaluate the line integral, where is the given curve
 16.16.61: Evaluate the line integral, where is the given curve
 16.16.62: Evaluate the line integral, where is the given curve
 16.16.63: Evaluate the line integral, where is the given curve
 16.16.64: Let be the vector field shown in the figure. F dr (a) If is the ver...
 16.16.65: The figure shows a vector field and two curves and . Are the line i...
 16.16.66: Evaluate the line integral , where is given by the vector function ...
 16.16.67: Evaluate the line integral , where is given by the vector function ...
 16.16.68: Evaluate the line integral , where is given by the vector function ...
 16.16.69: Evaluate the line integral , where is given by the vector function ...
 16.16.70: Use a calculator or CAS to evaluate the line integral correct to fo...
 16.16.71: Use a calculator or CAS to evaluate the line integral correct to fo...
 16.16.72: Use a calculator or CAS to evaluate the line integral correct to fo...
 16.16.73: Use a calculator or CAS to evaluate the line integral correct to fo...
 16.16.74: Use a graph of the vector field F and the curve C to guess whether ...
 16.16.75: Use a graph of the vector field F and the curve C to guess whether ...
 16.16.76: (a) Evaluate the line integral , where and is given by , . ; (b) Il...
 16.16.77: a) Evaluate the line integral , where and is given by , . ; (b) Ill...
 16.16.78: Find the exact value of , where is the curve with parametric equations
 16.16.79: (a) Find the work done by the force field on a particle that moves ...
 16.16.80: A thin wire is bent into the shape of a semicircle , . If the linea...
 16.16.81: A thin wire has the shape of the firstquadrant part of the circle ...
 16.16.82: (a) Write the formulas similar to Equations 4 for the center of mas...
 16.16.83: (b) Find the center of mass of a wire in the shape of the helix Fxk...
 16.16.84: Find the mass and center of mass of a wire in the shape of the heli...
 16.16.85: If a wire with linear density lies along a plane curve its moments ...
 16.16.86: If a wire with linear density lies along a space curve , its moment...
 16.16.87: Find the work done by the force field in moving an object along an ...
 16.16.88: Find the work done by the force field on a particle that moves alon...
 16.16.89: Find the work done by the force field on a particle that moves alon...
 16.16.90: The force exerted by an electric charge at the origin on a charged ...
 16.16.91: A 160lb man carries a 25lb can of paint up a helical staircase th...
 16.16.92: Suppose there is a hole in the can of paint in Exercise 43 and 9 lb...
 16.16.93: (a) Show that a constant force field does zero work on a particle t...
 16.16.94: (b) Is this also true for a force field , where is a constant and x...
 16.16.95: The base of a circular fence with radius 10 m is given by . The hei...
 16.16.96: An object moves along the curve shown in the figure from (1, 2) to ...
 16.16.97: Experiments show that a steady current in a long wire produces a ma...
 16.16.98: Find the work done by the gravitational field in moving a particle ...
 16.16.99: Determine whether or not the vector field is conservative.
 16.16.100: Determine whether or not the vector field is conservative
 16.16.101: (a) If , find a function such that . (b) Evaluate the line integral...
 16.16.102: If , find a function such that f F
 16.16.103: The figure shows a curve and a contour map of a function cos y j wh...
 16.16.104: A table of values of a function with continuous gradient is given. ...
 16.16.105: Determine whether or not is a conservative vector field. If it is, ...
 16.16.106: Determine whether or not is a conservative vector field. If it is, ...
 16.16.107: Determine whether or not is a conservative vector field. If it is, ...
 16.16.108: Determine whether or not is a conservative vector field. If it is, ...
 16.16.109: Determine whether or not is a conservative vector field. If it is, ...
 16.16.110: Determine whether or not is a conservative vector field. If it is, ...
 16.16.111: Determine whether or not is a conservative vector field. If it is, ...
 16.16.112: Determine whether or not is a conservative vector field. If it is, ...
 16.16.113: The figure shows the vector field and three curves that start at (1...
 16.16.114: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.115: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.116: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.117: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.118: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.119: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.120: (a) Find a function such that and (b) use F f fx, ysinx 2yC1 part (...
 16.16.121: Show that the line integral is independent of path and evaluate the...
 16.16.122: Show that the line integral is independent of path and evaluate the...
 16.16.123: Find the work done by the force field in moving an object from Pto ...
 16.16.124: Find the work done by the force field in moving an object from Pto ...
 16.16.125: Is the vector field shown in the figure conservative? Explain.
 16.16.126: Is the vector field shown in the figure conservative? Explain.
 16.16.127: If , use a plot to guess whether is conservative. Then determine wh...
 16.16.128: Let , where . Find curves and that are not closed and satisfy the e...
 16.16.129: Show that if the vector field is conservative and , , have continuo...
 16.16.130: Use Exercise 27 to show that the line integral is not independent o...
 16.16.131: Determine whether or not the given set is (a) open, (b) connected, ...
 16.16.132: Determine whether or not the given set is (a) open, (b) connected, ...
 16.16.133: Determine whether or not the given set is (a) open, (b) connected, ...
 16.16.134: Determine whether or not the given set is (a) open, (b) connected, ...
 16.16.135: Let . (a) Show that . (b) Show that is not independent of path. [Hi...
 16.16.136: (a) Suppose that is an inverse square force field, that is, for som...
 16.16.137: Evaluate , where is the triangular curve consisting of the line seg...
 16.16.138: Evaluate , where is the circle x 2 y 2 9
 16.16.139: Find the area enclosed by the ellipse x 2 a2 y 2 b2 1
 16.16.140: Evaluate , where is the boundary of the semiannular region in the u...
 16.16.141: If , show that for every positively oriented simple closed path tha...
 16.16.142: Evaluate the line integral by two methods: (a) directly and sin y d...
 16.16.143: Evaluate the line integral by two methods: (a) directly and sin y d...
 16.16.144: Evaluate the line integral by two methods: (a) directly and sin y d...
 16.16.145: Evaluate the line integral by two methods: (a) directly and sin y d...
 16.16.146: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.147: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.148: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.149: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.150: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.151: Use Greens Theorem to evaluate the line integral along the given po...
 16.16.152: Use Greens Theorem to evaluate . (Check the orientation of the curv...
 16.16.153: Use Greens Theorem to evaluate . (Check the orientation of the curv...
 16.16.154: Use Greens Theorem to evaluate . (Check the orientation of the curv...
 16.16.155: Use Greens Theorem to evaluate . (Check the orientation of the curv...
 16.16.156: Verify Greens Theorem by using a computer algebra system to evaluat...
 16.16.157: Verify Greens Theorem by using a computer algebra system to evaluat...
 16.16.158: Use Greens Theorem to find the work done by the force in moving a p...
 16.16.159: A particle starts at the point , moves along the axis to , and the...
 16.16.160: Use one of the formulas in (5) to find the area under one arch of t...
 16.16.161: If a circle with radius 1 rolls along the outside of the circle , a...
 16.16.162: (a) If is the line segment connecting the point to the point , show...
 16.16.163: Let be a region bounded by a simple closed path in the plane. Use ...
 16.16.164: Use Exercise 22 to find the centroid of a quartercircular region o...
 16.16.165: Use Exercise 22 to find the centroid of the triangle with vertices ...
 16.16.166: A plane lamina with constant density occupies a region in the plan...
 16.16.167: Use Exercise 25 to find the moment of inertia of a circular disk of...
 16.16.168: If is the vector field of Example 5, show that for every simple clo...
 16.16.169: Complete the proof of the special case of Greens Theorem by proving...
 16.16.170: Use Greens Theorem to prove the change of variables formula for a d...
 16.16.171: Show that the vector field is not conservative
 16.16.172: (a) Show that is a conservative vector field. (b) Find a function s...
 16.16.173: Show that the vector field cant be written as the curl of another v...
 16.16.174: Find (a) the curl and (b) the divergence of the vector field.
 16.16.175: Find (a) the curl and (b) the divergence of the vector field.
 16.16.176: Find (a) the curl and (b) the divergence of the vector field.
 16.16.177: Find (a) the curl and (b) the divergence of the vector field.
 16.16.178: Find (a) the curl and (b) the divergence of the vector field.
 16.16.179: Find (a) the curl and (b) the divergence of the vector field.
 16.16.180: Find (a) the curl and (b) the divergence of the vector field.
 16.16.181: Find (a) the curl and (b) the divergence of the vector field.
 16.16.182: The vector field F is shown in the xyplane and looks the same in a...
 16.16.183: The vector field F is shown in the xyplane and looks the same in a...
 16.16.184: The vector field F is shown in the xyplane and looks the same in a...
 16.16.185: Let be a scalar field and a vector field. State whether each expres...
 16.16.186: Determine whether or not the vector field is conservative. If it is...
 16.16.187: Determine whether or not the vector field is conservative. If it is...
 16.16.188: Determine whether or not the vector field is conservative. If it is...
 16.16.189: Determine whether or not the vector field is conservative. If it is...
 16.16.190: Determine whether or not the vector field is conservative. If it is...
 16.16.191: Determine whether or not the vector field is conservative. If it is...
 16.16.192: Is there a vector field on such that ? Explain.
 16.16.193: Is there a vector field on such that ? Explain.
 16.16.194: Show that any vector field of the form where , , are differentiable...
 16.16.195: Show that any vector field of the form is incompressible.
 16.16.196: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.197: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.198: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.199: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.200: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.201: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.202: Prove the identity, assuming that the appropriate partial derivativ...
 16.16.203: Verify each identity. (a) (b) (c) 2 r 3 12r
 16.16.204: Verify each identity. (a) (b) (c) 2 r 3 12r
 16.16.205: Verify each identity. (a) (b) (c) 2 r 3 12r
 16.16.206: Use Greens Theorem in the form of Equation 13 to prove Greens first...
 16.16.207: Use Greens first identity (Exercise 33) to prove Greens second iden...
 16.16.208: Recall from Section 14.3 that a function is called harmonic on if i...
 16.16.209: Use Greens first identity to show that if is harmonic on and if on ...
 16.16.210: This exercise demonstrates a connection between the curl vector and...
 16.16.211: Maxwells equations relating the electric field and magnetic field a...
 16.16.212: We have seen that all vector fields of the form satisfy the equatio...
 16.16.213: Identify and sketch the surface with vector equation
 16.16.214: Use a computer algebra system to graph the surface
 16.16.215: Find a vector function that represents the plane that passes throug...
 16.16.216: Find a parametric representation of the sphere
 16.16.217: Find a parametric representation for the cylinder
 16.16.218: Find a vector function that represents the elliptic paraboloid
 16.16.219: Find a parametric representation for the surface , that is, the top...
 16.16.220: Find parametric equations for the surface generated by rotating the...
 16.16.221: Find the tangent plane to the surface with parametric equations , ,...
 16.16.222: Find the surface area of a sphere of radius .
 16.16.223: Find the area of the part of the paraboloid that lies under the plane
 16.16.224: Determine whether the points and lie on the given surface
 16.16.225: Determine whether the points and lie on the given surface
 16.16.226: Identify the surface with the given vector equation
 16.16.227: Identify the surface with the given vector equation
 16.16.228: Identify the surface with the given vector equation
 16.16.229: Identify the surface with the given vector equation
 16.16.230: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.231: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.232: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.233: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.234: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.235: Use a computer to graph the parametric surface. Get a printout and ...
 16.16.236: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.237: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.238: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.239: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.240: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.241: Match the equations with the graphs labeled IVI and give reasons fo...
 16.16.242: Find a parametric representation for the surface.
 16.16.243: Find a parametric representation for the surface.
 16.16.244: Find a parametric representation for the surface.
 16.16.245: Find a parametric representation for the surface.
 16.16.246: Find a parametric representation for the surface.
 16.16.247: Find a parametric representation for the surface.
 16.16.248: Find a parametric representation for the surface.
 16.16.249: Find a parametric representation for the surface.
 16.16.250: Use a computer algebra system to produce a graph thatlooks like the...
 16.16.251: Use a computer algebra system to produce a graph that looks like th...
 16.16.252: Find parametric equations for the surface obtained by rotating the ...
 16.16.253: Find parametric equations for the surface obtained by rotating the ...
 16.16.254: a) What happens to the spiral tube in Example 2 (see Figure 5) if w...
 16.16.255: a) What happens to the spiral tube in Example 2 (see Figure 5) if w...
 16.16.256: Find an equation of the tangent plane to the given parametric surfa...
 16.16.257: Find an equation of the tangent plane to the given parametric surfa...
 16.16.258: Find an equation of the tangent plane to the given parametric surfa...
 16.16.259: Find an equation of the tangent plane to the given parametric surfa...
 16.16.260: Find the area of the surface
 16.16.261: Find the area of the surface
 16.16.262: Find the area of the surface
 16.16.263: Find the area of the surface
 16.16.264: Find the area of the surface
 16.16.265: Find the area of the surface
 16.16.266: Find the area of the surface
 16.16.267: Find the area of the surface
 16.16.268: Find the area of the surface
 16.16.269: Find the area of the surface
 16.16.270: Find the area of the surface
 16.16.271: Find the area of the surface correct to four decimal places by expr...
 16.16.272: Find the area of the surface correct to four decimal places by expr...
 16.16.273: Find, to four decimal places, the area of the part of the surface t...
 16.16.274: a) Use the Midpoint Rule for double integrals (see Section 15.1) wi...
 16.16.275: Find the area of the surface with vector equation , , . State your ...
 16.16.276: Find the exact area of the surface
 16.16.277: (a) Set up, but do not evaluate, a double integral for the area of ...
 16.16.278: (a) Show that the parametric equations , , , , , represent an ellip...
 16.16.279: (a) Show that the parametric equations , , , represent a hyperboloi...
 16.16.280: Find the area of the part of the sphere that lies inside the parabo...
 16.16.281: The figure shows the surface created when the cylinder intersects t...
 16.16.282: Find the area of the part of the sphere that lies inside the cylinder
 16.16.283: (a) Find a parametric representation for the torus obtained by rota...
 16.16.284: Compute the surface integral , where is the unit sphere
 16.16.285: Evaluate , where is the surface , , . (See Figure 2.)
 16.16.286: Evaluate , where is the surface whose sides are given by the cylind...
 16.16.287: Find the flux of the vector field across the unit
 16.16.288: Evaluate , where and is the boundary of the solid region enclosed b...
 16.16.289: The temperature in a metal ball is proportional to the square of th...
 16.16.290: Let be the boundary surface of the box enclosed by the y dS planes ...
 16.16.291: A surface consists of the cylinder , , together with its top and bo...
 16.16.292: Let be the hemisphere , and suppose is a continuous function with ,...
 16.16.293: Suppose that , where is a function of one variable such that . Eval...
 16.16.294: Evaluate the surface integral.
 16.16.295: Evaluate the surface integral.
 16.16.296: Evaluate the surface integral.
 16.16.297: Evaluate the surface integral.
 16.16.298: Evaluate the surface integral.
 16.16.299: Evaluate the surface integral.
 16.16.300: Evaluate the surface integral.
 16.16.301: Evaluate the surface integral.
 16.16.302: Evaluate the surface integral.
 16.16.303: Evaluate the surface integral.
 16.16.304: Evaluate the surface integral.
 16.16.305: Evaluate the surface integral.
 16.16.306: Evaluate the surface integral.
 16.16.307: Evaluate the surface integral.
 16.16.308: Evaluate the surface integral for the given vector field and the or...
 16.16.309: Evaluate the surface integral for the given vector field and the or...
 16.16.310: Evaluate the surface integral for the given vector field and the or...
 16.16.311: Evaluate the surface integral for the given vector field and the or...
 16.16.312: Evaluate the surface integral for the given vector field and the or...
 16.16.313: Evaluate the surface integral for the given vector field and the or...
 16.16.314: Evaluate the surface integral for the given vector field and the or...
 16.16.315: Evaluate the surface integral for the given vector field and the or...
 16.16.316: Evaluate the surface integral for the given vector field and the or...
 16.16.317: Evaluate the surface integral for the given vector field and the or...
 16.16.318: Evaluate the surface integral for the given vector field and the or...
 16.16.319: Evaluate the surface integral for the given vector field and the or...
 16.16.320: Evaluate correct to four decimal places, where is the surface
 16.16.321: Find the exact value of , where is the surface in Exercise 31.
 16.16.322: Find the value of correct to four decimal places, where is the part...
 16.16.323: Find the flux of across the part of the cylinder that lies above th...
 16.16.324: Find a formula for similar to Formula 10 for the case where is give...
 16.16.325: Find a formula for similar to Formula 10 for the case where is give...
 16.16.326: Find the center of mass of the hemisphere , if it has constant dens...
 16.16.327: Find the mass of a thin funnel in the shape of a cone , , if its de...
 16.16.328: (a) Give an integral expression for the moment of inertia about the...
 16.16.329: Let be the part of the sphere that lies above the plane . If has co...
 16.16.330: A fluid has density and flows with velocity , where and are measure...
 16.16.331: Seawater has density and flows in a velocity field , where and are ...
 16.16.332: Use Gausss Law to find the charge contained in the solid hemisphere...
 16.16.333: Use Gausss Law to find the charge contained in the solid hemisphere...
 16.16.334: The temperature at the point in a substance with conductivity is . ...
 16.16.335: The temperature at a point in a ball with conductivity is inversely...
 16.16.336: The temperature at a point in a ball with conductivity is inversely...
 16.16.337: Evaluate , where and is the curve of intersection of the plane and ...
 16.16.338: Use Stokes Theorem to compute the integral , where and is the part ...
 16.16.339: A hemisphere and a portion of a paraboloid are shown. Fx, y, zxy i ...
 16.16.340: Use Stokes Theorem to evaluate
 16.16.341: Use Stokes Theorem to evaluate
 16.16.342: Use Stokes Theorem to evaluate
 16.16.343: Use Stokes Theorem to evaluate
 16.16.344: Use Stokes Theorem to evaluate
 16.16.345: Use Stokes Theorem to evaluate . In each case is oriented countercl...
 16.16.346: Use Stokes Theorem to evaluate . In each case is oriented countercl...
 16.16.347: Use Stokes Theorem to evaluate . In each case is oriented countercl...
 16.16.348: Use Stokes Theorem to evaluate . In each case is oriented countercl...
 16.16.349: a) Use Stokes Theorem to evaluate , where and is the curve of inter...
 16.16.350: a) Use Stokes Theorem to evaluate , where and is the curve of inter...
 16.16.351: Verify that Stokes Theorem is true for the given vector field and s...
 16.16.352: Verify that Stokes Theorem is true for the given vector field and s...
 16.16.353: Verify that Stokes Theorem is true for the given vector field and s...
 16.16.354: Let be a simple closed smooth curve that lies in the plane . Show t...
 16.16.355: A particle moves along line segments from the origin to the points ...
 16.16.356: Evaluate CS where is the curve , . [Hint: Observe that lies on the ...
 16.16.357: If is a sphere and satisfies the hypotheses of Stokes Theorem, show...
 16.16.358: Suppose and satisfy the hypotheses of Stokes Theorem and , have con...
 16.16.359: Find the flux of the vector field over the unit sphere
 16.16.360: Evaluate , where and is the surface of the region bounded by the pa...
 16.16.361: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.362: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.363: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.364: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.365: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.366: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.367: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.368: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.369: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.370: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.371: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.372: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.373: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.374: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.375: Verify that the Divergence Theorem is true for the vector field 7. ...
 16.16.376: Use a computer algebra system to plot the vector field in the cube ...
 16.16.377: Use the Divergence Theorem to evaluate , where and is the top half ...
 16.16.378: Let . Find the flux of across the part of the paraboloid that lies ...
 16.16.379: A vector field is shown. Use the interpretation of divergence deriv...
 16.16.380: (a) Are the points and sources or sinks for the vector field shown ...
 16.16.381: Plot the vector field and guess where and where . Then calculate to...
 16.16.382: Plot the vector field and guess where and where . Then calculate to...
 16.16.383: Verify that for the electric field
 16.16.384: Use the Divergence Theorem to evaluate where is the sphere
 16.16.385: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.386: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.387: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.388: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.389: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.390: Prove each identity, assuming that and satisfy the conditions of th...
 16.16.391: Suppose and satisfy the conditions of the Divergence Theorem and is...
 16.16.392: A solid occupies a region with surface and is immersed in a liquid ...
 16.16.393: A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow...
 16.16.394: Gravel is being dumped from a conveyor belt at a rate of 30 , and i...
 16.16.395: A kite 100 ft above the ground moves horizontally at a speed of 8 f...
 16.16.396: Two sides of a triangle are 4 m and 5 m in length and the angle bet...
 16.16.397: How fast is the angle between the ladder and the ground changing in...
 16.16.398: Boyles Law states that when a sample of gas is compressed at a cons...
 16.16.399: When air expands adiabatically (without gaining or losing heat), it...
 16.16.400: If two resistors with resistances and are connected in parallel, as...
 16.16.401: Brain weight as a function of body weight in fish has been modeled ...
 16.16.402: Two sides of a triangle have lengths 12 m and 15 m. The angle betwe...
 16.16.403: Two carts, A and B, are connected by a rope 39 ft long that passes ...
 16.16.404: A television camera is positioned 4000 ft from the base of a rocket...
 16.16.405: A lighthouse is located on a small island 3 km away from the neares...
 16.16.406: A plane flies horizontally at an altitude of and passes directly ov...
 16.16.407: A Ferris wheel with a radius of is rotating at a rate of one revolu...
 16.16.408: A plane flying with a constant speed of 300 km!h passes over a grou...
 16.16.409: Two people start from the same point. One walks east at 3 mi!h and ...
 16.16.410: A runner sprints around a circular track of radius 100 m at a const...
 16.16.411: The minute hand on a watch is 8 mm long and the hour hand is 4 mm l...
 16.16.412: Find the linearization of the function at and use it to approximate...
 16.16.413: For what values of is the linear approximation
 16.16.414: Compare the values of and if and changes (a) from 2 to 2.05 and (b)...
 16.16.415: The radius of a sphere was measured and found to be 21 cm with a po...
 16.16.416: Find the linearization of the function at a f !x"!x a !%1
 16.16.417: Find the linearization of the function at a f !x"!ln x a !1
 16.16.418: Find the linearization of the function at a f !x"!cos x a !&$2
 16.16.419: Find the linearization of the function at a f !x"!x a !16
 16.16.420: Find the linear approximation of the function at and use it to appr...
 16.16.421: Find the linear approximation of the function at and use it to appr...
 16.16.422: Verify the given linear approximation at . Then determine the value...
 16.16.423: Verify the given linear approximation at . Then determine the value...
 16.16.424: Verify the given linear approximation at . Then determine the value...
 16.16.425: Verify the given linear approximation at . Then determine the value...
 16.16.426: Find the differential of each function.
 16.16.427: Find the differential of each function.
 16.16.428: Find the differential of each function.
 16.16.429: Find the differential of each function.
 16.16.430: (a) Find the differential and (b) evaluate for the given values of ...
 16.16.431: (a) Find the differential and (b) evaluate for the given values of ...
 16.16.432: (a) Find the differential and (b) evaluate for the given values of ...
 16.16.433: (a) Find the differential and (b) evaluate for the given values of ...
 16.16.434: Compute and for the given values of and . Then sketch a diagram lik...
 16.16.435: Compute and for the given values of and . Then sketch a diagram lik...
 16.16.436: Compute and for the given values of and . Then sketch a diagram lik...
 16.16.437: Compute and for the given values of and . Then sketch a diagram lik...
 16.16.438: Use a linear approximation (or differentials) to estimate the given...
 16.16.439: Use a linear approximation (or differentials) to estimate the given...
 16.16.440: Use a linear approximation (or differentials) to estimate the given...
 16.16.441: Use a linear approximation (or differentials) to estimate the given...
 16.16.442: Use a linear approximation (or differentials) to estimate the given...
 16.16.443: Use a linear approximation (or differentials) to estimate the given...
 16.16.444: Explain, in terms of linear approximations or differentials, why th...
 16.16.445: Explain, in terms of linear approximations or differentials, why th...
 16.16.446: Explain, in terms of linear approximations or differentials, why th...
 16.16.447: Trace or copy the graph of the function. Then sketch a graph of its...
 16.16.448: Trace or copy the graph of the function. Then sketch a graph of its...
 16.16.449: (a) If , use the definition of a derivative to find . (b) Find the ...
 16.16.450: (a) Find the asymptotes of the graph of and use them to sketch the ...
 16.16.451: The graph of is shown. State, with reasons, the numbers at which is...
 16.16.452: The figure shows the graphs of , , and . Identify each curve, and e...
 16.16.453: Let be the total value of US currency (coins and banknotes) in circ...
 16.16.454: The total fertility rate at time t, denoted by , is an estimate of ...
 16.16.455: Suppose that for all , where . Find .
 16.16.456: Let . (a) For what values of does exist? (b) At what numbers is f d...
 16.16.457: Evaluate lim xl 0 s 3 1 !cx "1 x, where c is a nonzero constant.
 16.16.458: Evaluate lim xl1 s 3 x "1 sx "1
 16.16.459: Find numbers a and b such that lim . xl0 sax !b "2 x !
 16.16.460: Evaluate lim xl0 $2x "1 $"$2x !1 $ x
 16.16.461: The figure shows a point P on the parabola and the point Q where th...
 16.16.462: If denotes the greatest integer function, find lim xl' x (x
 16.16.463: Sketch the region in the plane defined by each of the following equ...
 16.16.464: Find all values of a such that is continuous on
 16.16.465: A fixed point of a function is a number in its domain such that . (...
 16.16.466: If limxla %f!x"!t!x"&!2 and limxla %f !x""t!x"&!1 find limxla %f !x...
 16.16.467: a) The figure shows an isosceles triangle with . The bisector of an...
 16.16.468: (a) If we start from latitude and proceed in a westerly direction, ...
 16.16.469: If is a differentiable function and , use the definition of a deriv...
 16.16.470: Suppose is a function that satisfies the equation for all real numb...
 16.16.471: Suppose is a function with the property that for all x. Show that ....
 16.16.472: Differentiate:(a)f!x"!1 x (b)y !s 3 x 2
 16.16.473: Find equations of the tangent line and normal line to the curve at ...
 16.16.474: Find the points on the curve y !x 4 "6x V 2 !4 where the tangent li...
 16.16.475: The equation of motion of a particle is ,s !2t s 3 "5t 2 !3t !4 whe...
 16.16.476: If , find and . Compare the graphs of and
 16.16.477: At what point on the curve is the tangent line parallel to the line ?
 16.16.478: (a) How is the number e defined? (b) Use a calculator to estimate t...
 16.16.479: (a) Sketch, by hand, the graph of the function , paying particular ...
 16.16.480: Determine whether the statement is true or false. If it is true, ex...
 16.16.481: Determine whether the statement is true or false. If it is true, ex...
 16.16.482: Determine whether the statement is true or false. If it is true, ex...
 16.16.483: Determine whether the statement is true or false. If it is true, ex...
 16.16.484: Determine whether the statement is true or false. If it is true, ex...
 16.16.485: Determine whether the statement is true or false. If it is true, ex...
 16.16.486: Determine whether the statement is true or false. If it is true, ex...
 16.16.487: Determine whether the statement is true or false. If it is true, ex...
 16.16.488: Determine whether the statement is true or false. If it is true, ex...
 16.16.489: Find the local and absolute extreme values of the function on f#f t...
 16.16.490: Find the local and absolute extreme values of the function on f#f t...
 16.16.491: Find the local and absolute extreme values of the function on f#f t...
 16.16.492: Find the local and absolute extreme values of the function on f#f t...
 16.16.493: Find the local and absolute extreme values of the function on f#f t...
 16.16.494: Find the local and absolute extreme values of the function on f#f t...
 16.16.495: Evaluate the limit.lim xl0 tan ,x ln!1 !x"
 16.16.496: Evaluate the limit.lim xl0 1 "cos x x 2 !x
 16.16.497: Evaluate the limit.lim xl0 e4x "1 "4x x2
 16.16.498: Evaluate the limit.lim xl( e4x "1 "4x x2
 16.16.499: Evaluate the limit.lim ln x xl( x3 e"x
 16.16.500: Evaluate the limit.lim xl0!x2 lim ln x
 16.16.501: Evaluate the limit.limx l 1!%x x "1 "1 ln x
 16.16.502: Evaluate the limit.lim x l!,$2""!tan x" cos x
 16.16.503: Sketch the graph of a function that satisfies the given conditions:
 16.16.504: Sketch the graph of a function that satisfies the given conditions:
 16.16.505: Sketch the graph of a function that satisfies the given conditions:
 16.16.506: The figure shows the graph of the derivative of a function . (a) On...
 16.16.507: Use the guidelines of Section 4.5 to sketch the curve
 16.16.508: Use the guidelines of Section 4.5 to sketch the curve
 16.16.509: Use the guidelines of Section 4.5 to sketch the curve y !x 4 "3x 3 ...
 16.16.510: Use the guidelines of Section 4.5 to sketch the curve y !1 1 "x 2
 16.16.511: Use the guidelines of Section 4.5 to sketch the curve y !1 x!x "3" 2
 16.16.512: Use the guidelines of Section 4.5 to sketch the curve !1 x 2 "1 !x ...
 16.16.513: Use the guidelines of Section 4.5 to sketch the curve
 16.16.514: Use the guidelines of Section 4.5 to sketch the curve y !s1 "x !s1 !
 16.16.515: Use the guidelines of Section 4.5 to sketch the curve y !xs2 !x
 16.16.516: Use the guidelines of Section 4.5 to sketch the curve
 16.16.517: Use the guidelines of Section 4.5 to sketch the curve y !sin2 x "2 ...
 16.16.518: Use the guidelines of Section 4.5 to sketch the curve y !4x "tan x,...
 16.16.519: Use the guidelines of Section 4.5 to sketch the curve y !sin"1 !1$x"
 16.16.520: Use the guidelines of Section 4.5 to sketch the curve y !e2x"x
 16.16.521: Use the guidelines of Section 4.5 to sketch the curve
 16.16.522: Use the guidelines of Section 4.5 to sketch the curve
 16.16.523: Produce graphs of that reveal all the important aspects of the curv...
 16.16.524: Produce graphs of that reveal all the important aspects of the curv...
 16.16.525: Produce graphs of that reveal all the important aspects of the curv...
 16.16.526: Produce graphs of that reveal all the important aspects of the curv...
 16.16.527: Graph in a viewing rectangle that shows all the main aspects of thi...
 16.16.528: (a) Graph the function . (b) Explain the shape of the graph by comp...
 16.16.529: Use the graphs of to estimate the coordinates of the maximum and m...
 16.16.530: Use the graphs of to estimate the coordinates of the maximum and m...
 16.16.531: Investigate the family of functions . What features do the members ...
 16.16.532: Investigate the family of functions . What happens to the maximum a...
 16.16.533: Show that the equation has exactly one real root.
 16.16.534: Suppose that is continuous on , and for all in . Show that .
 16.16.535: By applying the Mean Value Theorem to the function on the interval ...
 16.16.536: For what values of the constants and is a point of inflection of th...
 16.16.537: Let , where is twice differentiable for all , for all , and is conc...
 16.16.538: Find two positive integers such that the sum of the first number an...
 16.16.539: Show that the shortest distance from the point to the straight line...
 16.16.540: Find the point on the hyperbola that is closest to the point
 16.16.541: Find the smallest possible area of an isosceles triangle that is ci...
 16.16.542: Find the volume of the largest circular cone that can be inscribed ...
 16.16.543: In , lies on , , cm, and cm. Where should a point be chosen on so t...
 16.16.544: Solve Exercise 55 when cm
 16.16.545: The velocity of a wave of length in deep water is where and are kno...
 16.16.546: A metal storage tank with volume is to be constructed in the shape ...
 16.16.547: What is a vector field? Give three examples that have physical F me...
 16.16.548: (a) What is a conservative vector field? (b) What is a potential fu...
 16.16.549: (a) Write the definition of the line integral of a scalar function ...
 16.16.550: (a) Define the line integral of a vector field along a smooth curve...
 16.16.551: State the Fundamental Theorem for Line Integrals
 16.16.552: (a) What does it mean to say that is independent of path? (b) If yo...
 16.16.553: State Greens Theorem
 16.16.554: Write expressions for the area enclosed by a curve in terms of line...
 16.16.555: Suppose is a vector field on . (a) Define curl . (b) Define div .c)...
 16.16.556: If , how do you test to determine whether is conservative? What if ...
 16.16.557: (a) What is a parametric surface? What are its grid curves? (b) Wri...
 16.16.558: (a) Write the definition of the surface integral of a scalar functi...
 16.16.559: a) What is an oriented surface? Give an example of a nonorientable ...
 16.16.560: State Stokes Theorem.
 16.16.561: State the Divergence Theorem.
 16.16.562: In what ways are the Fundamental Theorem for Line Integrals, Greens...
 16.16.563: Determine whether the statement is true or false. If it is true, ex...
 16.16.564: Determine whether the statement is true or false. If it is true, ex...
 16.16.565: Determine whether the statement is true or false. If it is true, ex...
 16.16.566: Determine whether the statement is true or false. If it is true, ex...
 16.16.567: Determine whether the statement is true or false. If it is true, ex...
 16.16.568: Determine whether the statement is true or false. If it is true, ex...
 16.16.569: Determine whether the statement is true or false. If it is true, ex...
 16.16.570: Determine whether the statement is true or false. If it is true, ex...
 16.16.571: A vector field , a curve , and a point are shown. (a) Is positive, ...
 16.16.572: Evaluate the line integral.
 16.16.573: Evaluate the line integral.
 16.16.574: Evaluate the line integral.
 16.16.575: Evaluate the line integral.
 16.16.576: Evaluate the line integral.
 16.16.577: Evaluate the line integral.
 16.16.578: Evaluate the line integral.
 16.16.579: Evaluate the line integral.
 16.16.580: Find the work done by the force field in moving a particle from the...
 16.16.581: Show that is a conservative vector field. Then find a function such...
 16.16.582: Show that is a conservative vector field. Then find a function such...
 16.16.583: Show that is conservative and use this fact to evaluate along the g...
 16.16.584: Show that is conservative and use this fact to evaluate along the g...
 16.16.585: Verify that Greens Theorem is true for the line integral , where co...
 16.16.586: Use Greens Theorem to evaluate , where is the triangle with vertice...
 16.16.587: Use Greens Theorem to evaluate , where is the circle with countercl...
 16.16.588: Find curl and div if Fx, y, zex sin y i ey sin z j ez sin x k FF
 16.16.589: Show that there is no vector field such that
 16.16.590: Show that, under conditions to be stated on the vector fields and G...
 16.16.591: If is any piecewisesmooth simple closed plane curve and and are di...
 16.16.592: If and are twice differentiable functions, show that
 16.16.593: If is a harmonic function, that is, , show that the line integral i...
 16.16.594: a) Sketch the curve with parametric equations (b) Fin
 16.16.595: Find the area of the part of the surface that lies above the triang...
 16.16.596: a) Find an equation of the tangent plane at the point to the parame...
 16.16.597: Evaluate the surface integral.
 16.16.598: Evaluate the surface integral.
 16.16.599: Evaluate the surface integral.
 16.16.600: Evaluate the surface integral.
 16.16.601: Verify that Stokes Theorem is true for the vector field , where is ...
 16.16.602: Use Stokes Theorem to evaluate , where , is the part of the sphere ...
 16.16.603: Use Stokes Theorem to evaluate , where , and is the triangle with v...
 16.16.604: Use the Divergence Theorem to calculate the surface integral , wher...
 16.16.605: Verify that the Divergence Theorem is true for the vector field , w...
 16.16.606: Compute the outward flux of through the ellipsoid
 16.16.607: Let Fx, y, z3x 2 yz 3yi x 3 z 3xj x 3 y 2zk 4x 2 9y 2
 16.16.608: Let Evaluate , where is shown in the figure.
 16.16.609: Find , where and is the outwardly oriented surface shown in the fig...
 16.16.610: If the components of have continuous second partial derivatives and...
 16.16.611: If is a constant vector, , and is an oriented, smooth surface with ...
 16.16.612: Let be a smooth parametric surface and let be a point such that eac...
 16.16.613: Find the positively oriented simple closed curve for which the valu...
 16.16.614: Let be a simple closed piecewisesmooth space curve that lies in a ...
 16.16.615: Investigate the shape of the surface with parametric equations . St...
 16.16.616: Prove the following identity: F GF G G F F curl G G curl F z 1 2 z 0 z
 16.16.617: The figure depicts the sequence of events in each cylinder of a fou...
Solutions for Chapter 16: VECTOR CALCULUS
Full solutions for Calculus: Early Transcendentals  6th Edition
ISBN: 9780495011668
Solutions for Chapter 16: VECTOR CALCULUS
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 6. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780495011668. Chapter 16: VECTOR CALCULUS includes 617 full stepbystep solutions. Since 617 problems in chapter 16: VECTOR CALCULUS have been answered, more than 37093 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Basic logistic function
The function ƒ(x) = 1 / 1 + ex

Conjugate axis of a hyperbola
The line segment of length 2b that is perpendicular to the focal axis and has the center of the hyperbola as its midpoint

Conversion factor
A ratio equal to 1, used for unit conversion

Degree of a polynomial (function)
The largest exponent on the variable in any of the terms of the polynomial (function)

Divergence
A sequence or series diverges if it does not converge

Division algorithm for polynomials
Given ƒ(x), d(x) ? 0 there are unique polynomials q1x (quotient) and r1x(remainder) ƒ1x2 = d1x2q1x2 + r1x2 with with either r1x2 = 0 or degree of r(x) 6 degree of d1x2

Gaussian elimination
A method of solving a system of n linear equations in n unknowns.

Linear function
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers

Measure of spread
A measure that tells how widely distributed data are.

Normal distribution
A distribution of data shaped like the normal curve.

Parameter
See Parametric equations.

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Quadratic regression
A procedure for fitting a quadratic function to a set of data.

Radian
The measure of a central angle whose intercepted arc has a length equal to the circle’s radius.

Removable discontinuity at x = a
lim x:a ƒ(x) = limx:a+ ƒ(x) but either the common limit is not equal ƒ(a) to ƒ(a) or is not defined

Row echelon form
A matrix in which rows consisting of all 0’s occur only at the bottom of the matrix, the first nonzero entry in any row with nonzero entries is 1, and the leading 1’s move to the right as we move down the rows.

Summation notation
The series a nk=1ak, where n is a natural number ( or ?) is in summation notation and is read "the sum of ak from k = 1 to n(or infinity).” k is the index of summation, and ak is the kth term of the series

Term of a polynomial (function)
An expression of the form anxn in a polynomial (function).

Time plot
A line graph in which time is measured on the horizontal axis.

xcoordinate
The directed distance from the yaxis yzplane to a point in a plane (space), or the first number in an ordered pair (triple), pp. 12, 629.