- 13.1: A vector field , a curve , and a point are shown. (a) Is positive, ...
- 13.2: Evaluate the line integral.is the arc of the parabola from
- 13.3: Evaluate the line integral.xC yz cos x ds C x t y 3 cos t z 3 sin t...
- 13.4: Evaluate the line integral.is the ellipse with counterclockwise ori...
- 13.5: Evaluate the line integral.xC y C
- 13.6: Evaluate the line integral.xC sxy dx e y dy xz dz C rt t 4 i t 2 j ...
- 13.7: Evaluate the line integral.xC xy dx y 2 dy yz dz C 1, 0, 1 3, 4, 2 ...
- 13.8: Evaluate the line integral.xC F dr Fx, y x y i x 2 j C rt sin t i 1...
- 13.9: Evaluate the line integral.xC F dr Fx, y, z ez i xz j x y k C rt t ...
- 13.10: Find the work done by the force field in moving a particle from the...
- 13.11: Show that is a conservative vector field. Then find a function such...
- 13.12: Show that is a conservative vector field. Then find a function such...
- 13.13: Show that is conservative and use this fact to evaluate along the g...
- 13.14: Show that is conservative and use this fact to evaluate along the g...
- 13.15: Verify that Greens Theorem is true for the line integral , where co...
- 13.16: Use Greens Theorem to evaluate where is the triangle with vertices
- 13.17: Use Greens Theorem to evaluate , where is the circle with countercl...
- 13.18: Find curl and div if
- 13.19: Show that there is no vector field such tha
- 13.20: Show that, under conditions to be stated on the vector fields and ,...
- 13.21: If is any piecewise-smooth simple closed plane curve and and are di...
- 13.22: If and are twice differentiable functions, show that
- 13.23: If is a harmonic function, that is, , show that the line integral i...
- 13.24: (a) Sketch the curve with parametric equations (b) Find xC 2xe 2y d...
- 13.25: Find the area of the part of the surface that lies above the triang...
- 13.26: (a) Find an equation of the tangent plane at the point to the param...
- 13.27: Evaluate the surface integral.
- 13.28: Evaluate the surface integral.
- 13.29: Evaluate the surface integral.
- 13.30: Evaluate the surface integral.
- 13.31: Verify that Stokes Theorem is true for the vector field , where is ...
- 13.32: Use Stokes Theorem to evaluate , where , is the part of the sphere ...
- 13.33: Use Stokes Theorem to evaluate , where , and is the triangle with v...
- 13.34: Use the Divergence Theorem to calculate the surface integral , wher...
- 13.35: Verify that the Divergence Theorem is true for the vector field , w...
- 13.36: Compute the outward flux of through the ellipsoid
- 13.37: Find , where and is the outwardly oriented surface shown in the fig...
- 13.38: Let Evaluate , where is shown in the figure.
- 13.39: If is a constant vector, , and is an oriented, smooth surface with ...
- 13.40: If the components of have continuous second partial derivatives and...
Solutions for Chapter 13: Essential Calculus 2nd Edition
Full solutions for Essential Calculus | 2nd Edition
a(b + c) = ab + ac and related properties
Elementary row operations
The following three row operations: Multiply all elements of a row by a nonzero constant; interchange two rows; and add a multiple of one row to another row
Focal width of a parabola
The length of the chord through the focus and perpendicular to the axis.
Identity involving a trigonometric function of u/2.
Side opposite the right angle in a right triangle.
Instantaneous rate of change
See Derivative at x = a.
Inverse cosine function
The function y = cos-1 x
Inverse cotangent function
The function y = cot-1 x
A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative
Linear regression equation
Equation of a linear regression line
A degree 4 polynomial function.
Numbers that can be written as a/b, where a and b are integers, and b ? 0.
Real number line
A horizontal line that represents the set of real numbers.
Reflection through the origin
x, y and (-x,-y) are reflections of each other through the origin.
A 90° angle.
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the right-hand end point of each subinterval.
Ratio change in y/change in x
Sum of an infinite geometric series
Sn = a 1 - r , |r| 6 1
Vertex of a parabola
The point of intersection of a parabola and its line of symmetry.
The product of a force applied to an object over a given distance W = ƒFƒ ƒAB!ƒ.