?Find the work done by the force fieldF(x, y, z) = z i + x j + y kin moving a particle

What is a vector field? Give three examples that have physical meaning.

(a) What is a conservative vector field? (b) What is a potential function?

(a) Write the definition of the line integral of a scalar function f along a smooth curve C with respect to arc length. (b) How do you evaluate such a line integral? (c) Write expressions for the mass and center of mass of a thin wire shaped like a curve C if the wire has linear density function (x, y). (d) Write the definitions of the line integrals along C of a scalar function f with respect to x, y, and z. (e) How do you evaluate these line integrals?

(a) Define the line integral of a vector field \(\mathbf{F}\) along a smooth curve C given by a vector function \(\mathbf{r}(t)\). (b) If \(\mathbf{F}\) is a force field, what does this line integral represent? (c) If \(\mathbf{F}=\langle P, Q, R\rangle\), what is the connection between the line integral of \(\mathbf{F}\) and the line integrals of the component functions P, Q, and R ?

State the Fundamental Theorem for Line Integrals.

(a) What does it mean to say that ?C F dr is independent of path? (b) If you know that ?C F dr is independent of path, what can you say about F?

State Green’s Theorem.

Write expressions for the area enclosed by a curve C in terms of line integrals around C.

Suppose F is a vector field on ?3. (a) Define curl F. (b) Define div F. (c) If F is a velocity field in fluid flow, what are the physical interpretations of curl F and div F?

If \(\mathbf{F}=P \mathbf{i}+Q \mathbf{j}\), how do you determine whether \(\mathbf{F}\) is conservative? What if \(\mathbf{F}\) is a vector field on \(\mathbb{R}^3\) ?

(a) What is a parametric surface? What are its grid curves? (b) Write an expression for the area of a parametric surface. (c) What is the area of a surface given by an equation z = g(x, y)?

(a) Write the definition of the surface integral of a scalar function f over a surface S. (b) How do you evaluate such an integral if S is a parametric surface given by a vector function r(u, v)? (c) What if S is given by an equation z = g(x, y)? (d) If a thin sheet has the shape of a surface S, and the density at (x, y, z) is ????(x, y, z), write expressions for the mass and center of mass of the sheet.

(a) What is an oriented surface? Give an example of a nonorientable surface. (b) Define the surface integral (or flux) of a vector field F over an oriented surface S with unit normal vector n. (c) How do you evaluate such an integral if S is a parametric surface given by a vector function r(u, v)? (d) What if S is given by an equation z = g(x, y)?

State Stokes’ Theorem.

State the Divergence Theorem.

In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F is a vector field, then div F is a vector field.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F is a vector field, then curl F is a vector field.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f has continuous partial derivatives of all orders on \(\mathbb{R}^{3}\), then \(\text {div}(\text {curl}\ \nabla f)=0\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f has continuous partial derivatives on ?3 and C is any circle, then ?C = ?f dr = 0.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F = P i + Q j and Py = Qx in an open region D, then F is conservative.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. ??C f(x, y) ds = ??C f(x, y) ds

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F and G are vector fields and div F = div G, then F = G.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The work done by a conservative force field in moving a particle around a closed path is zero.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F and G are vector fields, then curl(F + G) = curl F + curl G

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If F and G are vector fields, then curl(F G) = curl F curl G

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If S is a sphere and F is a constant vector field, then ?S F dS = 0.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. There is a vector field F such that curl F = x i + y j + z k

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The area of the region bounded by the positively oriented, piecewise smooth, simple closed curve C is A = ?C y dx.

A vector field F, a curve C, and a point P are shown. (a) Is ?C F dr positive, negative, or zero? Explain. (b) Is div F(P) positive, negative, or zero? Explain.

Evaluate the line integral. ?C x ds, C is the arc of the parabola y = x2 from (0, 0) to (1, 1)

Evaluate the line integral. \(\int_{C} y z \cos x d s,\) C: x = t, y = 3 cos t, z = 3 sin t, \(0 \leqslant t \leqslant\)

Evaluate the line integral. ?C y dx + (x + y2) dy, C is the ellipse 4x2 + 9y2 = 36 with counterclockwise orientation

Evaluate the line integral. ?C y3 dx + x2 dy, C is the arc of the parabola x = 1 ? y2 from (0, ?1) to (0, 1)

Evaluate the line integral. ?C dx + ey dy 1 xz dz, C is given by r(t) = t4 i + t2 j + t3 k, 0 ? t ? 1

Evaluate the line integral. ?C xy dx + y2 dy + yz dz, C is the line segment from (1, 0, ?1), to (3, 4, 2)

Evaluate the line integral. ?C F dr, where F(x, y) = xy i + x2 j and C is given by r(t) = sin t i + (1 + t) j, 0 ? t ? ????

Evaluate the line integral. \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, \ y, \ z)=e^{z} \ \mathbf{i}+x z \ \mathbf{j}+(x+y) \ \mathbf{k}\) and C is given by \(\mathbf{r}(t)=t^{2} \ \mathbf{i}+t^{3} \ \mathbf{j}-t \ \mathbf{k}, \ 0 \leqslant t \leqslant 1\)

Find the work done by the force field F(x, y, z) = z i + x j + y k in moving a particle from the point (3, 0, 0) to the point (0, ????/2, 3) along each path. (a) A straight line (b) The helix x = 3 cos t, y = t, z = 3 sin t

Show that \(\mathbf{F}\) is a conservative vector field. Then find a function \(f\) such that \(\mathbf{F}=\nabla f\). \(\mathbf{F}(x, y)=(1+x y) e^{x y} \mathbf{i}+\left(e^{y}+x^{2} e^{x y}\right) \mathbf{j}\) Equation Transcription: Text Transcription: F F=nabla f F(x,y)=(1+xy)e^xy i+(e^y+x^2e^xy)j

Show that F is a conservative vector field. Then find a function f such that F = ?f. F(x, y, z) = sin y i + x cos y j ? sin z k

Show that F is conservative and use this fact to evaluate ?C F dr along the given curve. F(x, y) = (4x3y2 ? 2xy3) i + (2x4y ? 3x2y2 + 4y3) j, C: r(t) = (t + sin ????t) i + (2t 1 cos ????t) j, 0 ? t ? 1

Show that F is conservative and use this fact to evaluate ?C F dr along the given curve. F(x, y, z) = ey i + (xey + ez) j + yez k, C is the line segment from (0, 2, 0) to (4, 0, 3)

Verify that Green's Theorem is true for the line integral \(\int_C x y^2 d x-x^2 y d y\), where C consists of the parabola \(y=x^2\) from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1)

Use Green’s Theorem to evaluate ?C dx + 2xydy where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

Use Green's Theorem to evaluate \(\int_{c} x^{2} y \ d x-x y^{2} \ d y\), where C is the circle \(x^{2}+y^{2}=4\) with counterclockwise orientation.

Find curl F and div F if F(x, y, z) = e?x sin y i + e?y sin z j + e?z sin x k

Show that there is no vector field G such that curl G = 2x i + 3yz j ? xz2 k

If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl(F G) = F div G ? G div F + (G ?)F ? (F ?)G

If C is any piecewise-smooth simple closed plane curve and f and t are differentiable functions, show that ?C f(x) dx + g(y) dy = 0.

If f and g are twice differentiable functions, show that ?2(fg) = f?2g + g?2f + 2?f ?g

If f is a harmonic function, that is, ?2 f = 0, show that the line integral ? fydx ? fxdy is independent of path in any simple region D.

(a) Sketch the curve C with parametric equations x = cos t y = sin t z = sin t 0 ? t ? 2???? (b) Find ?C 2xe2y dx + (2x2e2y + 2y cot z)dy ? y2csc2z dz.

Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 2).

(a) Find an equation of the tangent plane at the point (4, -2, 1) to the parametric surface S given by \(\mathbf{r}(u,\ v)=v^2\mathbf{\ i}-uv\mathbf{\ j}+u^{2\ }\mathbf{k}\) \(0 \leqslant u \leqslant 3, \ -3 \leqslant v \leqslant 3\) (b) Graph the surface S and the tangent plane found in part (a). (c) Set up, but do not evaluate, an integral for the surface area of S. (d) If \(\mathbf{F}(x,\ y,\ z)=\frac{z^2}{1+x^2}\mathbf{\ i}+\frac{x^2}{1+y^2}\mathbf{\ j}+\frac{y^2}{1+z^2}\mathbf{\ k}\) use a computer algebra system to find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) correct to four decimal places.

Evaluate the surface integral. ?S z dS, where S is the part of the paraboloid z = x2 + y2 that lies under the plane z = 4

Evaluate the surface integral. ?S (x2z + y2z)dS, where S is the part of the plane z = 4 + x + y that lies inside the cylinder x2 + y2 = 4

Evaluate the surface integral. ?S F dS, where F(x, y, z) = xz i ? 2y j + 3x k and S is the sphere x2 + y2 + z2 = 4 with outward orientation

Evaluate the surface integral. ?S F dS, where F(x, y, z) = x2 i + xy j + z k and S is the part of the paraboloid z = x2 + y2 below the plane z = 1 with upward orientation

Verify that Stokes’ Theorem is true for the vector field F(x, y, z) = x2 i + y2 j + z2 k, where S is the part of the paraboloid z = 1 ? x2 ? y2 that lies above the xy-plane and S has upward orientation.

Use Stokes' Theorem to evaluate \(\iint_S\) curl \(\mathbf{F} \cdot d \mathbf{S}\), where \(\mathbf{F}(x, y, z)=x^2 y z \mathbf{i}+y z^2 \mathbf{j}+z^3 e^{x y} \mathbf{k}\), S is the part of the sphere \(x^2+y^2+z^2=5\) that lies above the plane z=1, and S is oriented upward.

Use Stokes' Theorem to evaluate , where and is the triangle with vertices , and , oriented counterclockwise as viewed from above.

Use the Divergence Theorem to calculate the surface integral \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\), where \(\mathbf{F}(x,\ y,\ z)=x^{3\ }\mathbf{i}+y^3\ \mathbf{j}+z^3\mathbf{\ k}\) and S is the surface of the solid bounded by the cylinder \(x^{2}+y^{2}=1\) and the planes \(z=0\) and \(z=2\).

Verify that the Divergence Theorem is true for the vector field , where is the unit ball .

Compute the outward flux of through the ellipsoid .

Let Evaluate , where is the curve with initial point and terminal point shown in the figure.

Let \(\mathbf{F}(x, y)=\frac{\left(2 x^3+2 x y^2-2 y\right) \mathbf{i}+\left(2 y^3+2 x^2 y+2 x\right) \mathbf{j}}{x^2+y^2}\) Evaluate \(\oint_C \mathbf{F} \cdot d \mathbf{r}\), where C is shown in the figure.

Find , where and is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed).

If the components of \(F\) have continuous second partial derivatives and \(S\) is the boundary surface of a simple solid region, show that \(\iint_{S} \operatorname{curl} F \cdot d S=0\) Equation Transcription: ?S Text Transcription: F S double integral _S curl F . dS = 0

If is a constant vector, , and is an oriented, smooth surface with a simple, closed, smooth, positively oriented boundary curve , show that

Let be a smooth parametric surface and let be a point such that each line that starts at intersects at most once. The solid angle subtended by at is the set of lines starting at and passing through . Let ) be the intersection of with the surface of the sphere with center and radius . Then the measure of the solid angle (in steradians) is defined to be Apply the Divergence Theorem to the part of between and to show that where is the radius vector from to any point on , and the unit normal vector is directed away from . This shows that the definition of the measure of a solid angle is independent of the radius of the sphere. Thus the measure of the solid angle is equal to the area subtended on a unit sphere. (Note the analogy with the definition of radian measure.) The total solid angle subtended by a sphere at its center is thus steradians.

Find the positively oriented simple closed curve \(C\) for which the value of the line integral \(\int_{C} \left(y^{3}-y\right) \ d x-2 x^{3} \ d y\) is a maximum.

Let be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector and has positive orientation with respect to . Show that the plane area enclosed by is

Investigate the shape of the surface with parametric equations \(x=\sin u, y=\sin v\), \(z=\sin (u+v)\). Start by graphing the surface from several points of view. Explain the appearance of the graphs by determining the traces in the horizontal planes \(Z=0, z=\pm 1\), and \(Z=\pm \frac{1}{2}\). Equation Transcription: Text Transcription: x=sin u, y=sin v z=sin(u+v) z=0, z=pm 1 z=pm 1/2

Prove the following identity: \(\nabla(F \cdot G)=(F \cdot \nabla) G+(G \cdot \nabla) F+F \times \operatorname{curl} G+G \times \operatorname{curl} F\) Equation Transcription: ??? Text Transcription: nabla (F . G)=(F . nabla)G+ (G . nabla)F + F x curl G + G x curl F

The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating crankshaft. Let P(t) and V(t) be the pressure and volume within a cylinder at time t, where \(a \leqslant t \leqslant b\) gives the time required for a complete cycle. The graph shows how P and V vary through one cycle of a four-stroke engine. During the intake stroke (from (1) to (2) a mixture of air and gasoline at atmospheric pressure is drawn into a cylinder through the intake valve as the piston moves downward. Then the piston rapidly compresses the mix with the valves closed in the compression stroke (from (2) to (3)) during which the pressure rises and the volume decreases. At (3) the sparkplug ignites the fuel, raising the temperature and pressure at almost constant volume to (4). Then, with valves closed, the rapid expansion forces the piston downward during the power stroke (from (4) to (5)). The exhaust valve opens, temperature and pressure drop, and mechanical energy stored in a rotating flywheel pushes the piston upward, forcing the waste products out of the exhaust valve in the exhaust stroke. The exhaust valve closes and the intake valve opens. We're now back at (1) and the cycle starts again. (a) Show that the work done on the piston during one cycle of a four-stroke engine is \(W=\int_{C} P \ d V\), where C is the curve in the PV-plane shown in the figure. [Hint: Let x(t) be the distance from the piston to the top of the cylinder and note that the force on the piston is \(\mathbf{F}=A P(t) \ \mathbf{i}\), where A is the area of the top of the piston. Then \(W=\int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}\), where \(C_{1}\) is given by \(\mathbf{r}(t)=x(t) \ \mathbf{i}, \ a \leqslant t \leqslant b\). An alternative approach is to work directly with Riemann sums.] (b) Use Formula 16.4.5 to show that the work is the difference of the areas enclosed by the two loops of C.

The set of all points within a perpendicular distance \(r\) from a smooth simple curve \(C\) in \(R^3 \) form a "tube," which we denote by Tube \((C,r)\); see the figure at the left. (We assume that \(r\) is small enough that the tube does not intersect itself.) It may seem that the volume of such a tube would depend on the twists and turns of \(C\), but in this problem you will find a formula for the volume of Tube \(C,r)\) which, perhaps surprisingly, depends only on \(r\) and the length of \(C\). We assume that \(C\) is parameterized with respect to arc length \(s\) as \(r(s)\), where \(a \leqslant s \leqslant b\) so the arc length of \(C\) is \(L=b-a\). (a) Show that the surface of Tube \((C, q)\) is parameterized by \(X(u, v)=r(u)+q \cos v N(u)+q \sin v B(u)\) \(\quad\) \(\quad\) \(\quad\) \(a \leqslant u \leqslant b, 0 \leqslant v \leqslant 2 \pi\) a. First choice b. 2nd choice c. 3rd choice d. None of the above where \(N\) and \(B\) are the unit normal and binormal vectors for \(C\). (b) Use the Frenet-Serret Formulas (Exercises 13.3.71-72) and the Pythagorean Theorem for vectors (Exercise 12.3.66) to show that \(\left|X_{u}(u, v) \times X_{v}(u, v)\right|=q[1-\kappa(u) q \cos v]\) and so the surface area of Tube \((C,q)\) is \(S(q)=\int_{a}^{b} \int_{0}^{2 \pi}\left|X_{u}(u, v) \times X_{v}(u, v)\right| d v d u=2 \pi q L\) (c) Consider a thin tubular shell of radius \(q\) and thickness \(\delta q\) along \(C\), a cross-section of which is shown in the figure. Observe that the volume of the shell is approximately \(\Delta q S(q)\) and conclude that the volume of Tube \((C, r)\) is \(\int_{0}^{r} S(q) d q=\pi r^{2} L\) (d) Find the volume of a tube of radius \(r=0.2\) around the helix \(r(t)=\langle\cos t, \sin t, t\rangle\), \(0 \leqslant t \leqslant 4 \pi\) (e) Find the volume of the torus in Example 8.3.7. Equation Transcription: ?? ???? ? ?? Text Transcription: r C R^3 (C, r) r C (C, r) r C C s r(s) a leqslant s leqslant b C L=b-a (C, q) X(u,v)=r(u)+q cos v N(u) + q sin v B(u) a leqslant u leqslant b, 0 leqslant v leqslant 2pi N B C |X_u (u,v) x X_v (u,v)| =q[1-k(u)q cos v] (C, q) S(q)=integral _a ^b integral _0 ^2pi |X_u (u,v) x X_v (u, v)| dvdu=2pi q L q nabla q C nabla q S(q) (C, r) integral _0 ^r S(q) dq=pi r^2 L r=0.2 r(t) = langle cos t, sin t, t rangle 0 leqslant t leqslant 4pi