In Exercises 51 and 52, use a computer algebra system to graphthe surface represented by

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=(2 y-z) \mathbf{i}+e^{z} \mathbf{j}+x y z \mathbf{k}\) Text Transcription: F(x,y,z)=(2y-z)i+e^zj+xyzk

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+x^{2} \mathbf{k}\) Text Transcription: F(x, y, z)=x^2i+y^2j+x^2k

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=2 z \mathbf{i}-4 x^{2} \mathbf{j}+\arctan x \mathbf{k}\) Text Transcription: F(x, y, z)=2zi-4x^2j+arctan xk

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=x \sin y \mathbf{i}-y \cos x \mathbf{j}+y z^{2} \mathbf{k}\) Text Transcription: F(x, y, z)=x sin yi-y cos xj+yz^2k

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=e^{x^{2}+y^{2}} \mathbf{i}+e^{y^{2}+z^{2}} \mathbf{j}+x y z \mathbf{k}\) Text Transcription: F(x, y, z)=e^x^2+y^2i+e^y^2+z^2j+xyzk

In Exercises 1-6, find the curl of the vector field F. \(\mathbf{F}(x, y, z)=\arcsin y \mathbf{i}+\sqrt{1-x^{2}} \mathbf{j}+y^{2} \mathbf{k}\) Text Transcription: F(x, y, z)=arcsin y i+sqrt 1-x^2 j+y^2k

In Exercises 7-10, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s=\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. F(x, y, z) = (-y + z)i + (x - z)j + (x - y)k \(S: z=9-x^{2}-y^{2}, \quad z \geq 0\) Text Transcription: int_C F cdot T ds=int_C F cdot dr S: z=9-x^2-y^2,z >= 0

In Exercises 7-10, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s=\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. F(x, y, z) = (-y + z)i + (x - z)j + (x - y)k \(S: z=\sqrt{1-x^{2}-y^{2}}\) Text Transcription: int_C F cdot T ds=int_C F cdot dr S: z=sqrt 1-x^2-y^2

In Exercises 7-10, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s=\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. F(x, y, z) = xyzi + yj + zk S: 6x + 6y + z = 12, first octant Text Transcription: int_C F cdot T ds=int_C F cdot dr

In Exercises 7-10, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s=\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+x^{2} \mathbf{j}+y^{2} \mathbf{k}\) \(S: z=y^{2}, \quad 0 \leq x \leq a, \quad 0 \leq y \leq a\) Text Transcription: int_C F cdot T ds=int_C F cdot dr F(x,y,z)=z^2i+x^2j+y^2k S: z=y^2, 0 <= x <= a, 0 <= y <= a

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + 3zj + xk C: triangle with vertices (2, 0, 0), (0, 2, 0), (0, 0, 2) Text Transcription: int_C F cdot dr

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=\arctan \frac{x}{y} \mathbf{i}+\ln \sqrt{x^{2}+y^{2}} \mathbf{j}+\mathbf{k}\) C : triangle with vertices (0, 0, 0), (1, 1, 1), (0, 0, 2) Text Transcription: int_C F cdot dr F(x,y,z)=arctan x/y i+ln sqrt x^2+y^2 j+k

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+2 x \mathbf{j}+y^{2} \mathbf{k}\) \(S: z=1-x^{2}-y^{2}, \quad z \geq 0\) Text Transcription: int_C F cdot dr F(x,y,z)=z^2i+2xj+y^2k S: z=1-x^2-y^2, z >= 0

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 4xzi + yj + 4xyk \(S: z=9-x^{2}-y^{2}, \quad z \geq 0\) Text Transcription: int_C F cdot dr S: z=9-x^2-y^2, z >= 0

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) \(S: z=\sqrt{4-x^{2}-y^{2}}\) Text Transcription: int_C F cdot dr F(x,y,z)=z^2i+yj+zk S: z=sqrt 4-x^2-y^2

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+z^{2} \mathbf{j}-x y z \mathbf{k}\) \(S: z=\sqrt{4-x^{2}-y^{2}}\) Text Transcription: int_C F cdot dr F(x, y, z)=x^2i+z^2j-x y z k S: z=sqrt 4-x^2-y^2

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=-\ln \sqrt{x^{2}+y^{2}} \mathbf{i}+\arctan \frac{x}{y} \mathbf{j}+\mathbf{k}\) S: z = 9 - 2x - 3y over \(r=2 \sin 2 \theta\) in the first octant Text Transcription: int_C F cdot dr F(x,y,z)=-ln sqrtx^2+y^2i + arctan x/y j +k r=2 sin 2 theta

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=y z \mathbf{i}+(2-3 y) \mathbf{j}+\left(x^{2}+y^{2}\right) \mathbf{k}, \quad x^{2}+y^{2} \leq 16\) S: the first-octant portion of \(x^{2}+z^{2}=16\) over \(x^{2}+y^{2}=16\) Text Transcription: int_C F cdot dr F(x,y,z)=yzi+(2-3y)j+(x^2+y^2)k, x^2+y^2<=16 x^2+z^2=16 x^2+y^2=16

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = xyzi + yj + zk \(S: z=x^{2}, \quad 0 \leq x \leq a, \quad 0 \leq y \leq a\) N is the downward unit normal to the surface. Text Transcription: int_C F cdot dr S:z=x^2, 0<=x<=a, 0<=y<=a

In Exercises 11-20, use Stokes's Theorem to evaluate \(\int_{C} F \cdot d r\). Use a computer algebra system to verify your results. In each case, C is oriented counterclockwise as viewed from above. F(x,y,z) = xyzi + yj + zk, \(x^{2}+y^{2} \leq a^{2}\) S: the first-octant portion of \(z=x^{2}\) over \(x^{2}+y^{2}=a^{2}\) Text Transcription: int_C F cdot dr x^2+y^2<=a^2 z=x^2 x^2+y^2=a^2

Motion of a Liquid In Exercises 21 and 22, the motion of a liquid in a cylindrical container of radius 1 is described by the velocity field F(x, y, z). Find \(\int_{S} \int(\operatorname{curl} \mathrm{F}) \cdot \mathrm{N} d \mathrm{~S}\), where S is the upper surface of the cylindrical container. F(x, y, z) = i + j - 2k Text Transcription: int_S int (curl F) cdot N dS

Motion of a Liquid In Exercises 21 and 22, the motion of a liquid in a cylindrical container of radius 1 is described by the velocity field F(x, y, z). Find \(\int_{S} \int(\operatorname{curl} \mathrm{F}) \cdot \mathrm{N} d \mathrm{~S}\), where S is the upper surface of the cylindrical container. F(x, y, z) = -zi + yk Text Transcription: int_S int (curl F) cdot N dS

State Stokes's Theorem.

Give a physical interpretation of curl.

Let f and g be scalar functions with continuous partial derivatives, and let C and S satisfy the conditions of Stokes's Theorem. Verify each identity. (a) \(\int_{C}(f \nabla g) \cdot d \mathbf{r}=\int_{S} \int(\nabla f \times \nabla g) \cdot \mathbf{N} d S\) (b) \(\int_{C}(f \nabla f) \cdot d \mathbf{r}=0\) (c) \(\int_{C}(f \nabla g+g \nabla f) \cdot d \mathbf{r}=0\) Text Transcription: int_C(f nabla g) cdot d r=int_S int(nabla f x nabla g) cdot N dS int_C(f nabla f) cdot d r=0 int_C(f nabla g+g nabla f) cdot d r=0

Demonstrate the results of Exercise 25 for the functions f(x, y, z) = xyz and g(x, y, z) = z. Let S be the hemisphere \(z=\sqrt{4-x^{2}-y^{2}}\). Text Transcription: z=sqrt 4-x^2-y^2

Let C be a constant vector. Let S be an oriented surface with a unit normal vector N, bounded by a smooth curve C. Prove that \(\int_{S} \int \mathbf{C} \cdot \mathbf{N} d S=\frac{1}{2} \int_{C}(\mathbf{C} \times \mathbf{r}) \cdot d \mathbf{r}\). Text Transcription: int_S int C cdot N dS=1/2 int_C (C x r) cdot dr

Verify Stokes's Theorem for each given vector field and upward oriented surface. Is the line integral or the double integral easier to set up? to evaluate? Explain. (a) \(\mathbf{F}(x, y, z)=e^{y+z} \mathbf{i}\) C: square with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0) (b) \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+x^{2} \mathbf{j}+y^{2} \mathbf{k}\) S : the portion of the paraboloid \(z=x^{2}+y^{2}\) that lies below the plane z = 4 Text Transcription: F(x,y,z)=e^y+zi F(x,y,z)=z^2i+x^2j+y^2k z=x^2+y^2

Let \(\mathbf{G}(x, y)=\left(\frac{-y}{x^{2}+4 y^{2}}, \frac{x}{x^{2}+4 y^{2}}, 0\right)\). Prove or disprove that there is a vector-valued function F(x, y, z) = (M(x, y, z), N(x, y, z), P(x, y, z)) with the following properties. (i) M, N, P have continuous partial derivatives for all \((x, y, z) \neq(0,0,0)\); (ii) Curl F = 0 for all \((x, y, z) \neq(0,0,0)\); (iii) F(x, y, 0) = G(x, y). Text Transcription: G(x,y)=(-y/x^2+4y^2, x/x^2+4y^2, 0) (x, y, z) not= (0,0,0)

Let Prove or disprove that there is a vector-valued function with the following properties. (i) have continuous partial derivatives for all (ii) Curl for all (iii) This problem was composed by the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. Fx, y, 0  Gx, y

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). F(x, y) = xyi + 2xyj \(C: \mathbf{r}(t)=t^{2} \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1\) Text Transcription: int_C F cdot dr C:r(t)=t^2i+t^2j, 0<=t<=1

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). F(x, y) = (x - y)i + (x+y)j C: r(t) = 4 cos ti + 3 sin tj, \(0 \leq t \leq 2 \pi\) Text Transcription: int_C F cdot dr 0 <= t <= 2 pi

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). F(x, y, z) = xi + yj + zk C:r(t) = 2 cos ti + 2 sin tj + tk, \(0 \leq t \leq 2 \pi\) Text Transcription: int_C F cdot dr 0 <= t <= 2 pi

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). F(x, y, z) = (2y - z)i + (z - x)j + (x - y)k C: curve of intersection of \(x^2+z^2=4\) and \(y^2+z^2=4\) from (2, 2, 0) to (0, 0, 2) Text Transcription: int_C F cdot dr x^2+z^2=4 y^2+z^2=4

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). F(x, y, z) = (y + z)i + (x + z)j + (x + y)k C: curve of intersection of \(z=x^2+y^2\) and y = x from (0, 0, 0) to (2, 2, 8) Text Transcription: int_C F cdot dr z=x^2+y^2

In Exercises 31-36, evaluate \(\int_{C} F \cdot d r\). \(\mathbf F(x,y,z)=(x^2-z)\mathbf i + (y^2+z)\mathbf j+ x\mathbf k\) C: curve of intersection of \(z=x^2\) and \(x^2+y^2=4\) from (0, -2, 0) to (0, 2, 0) Text Transcription: int_C F cdot dr F(x,y,z)=(x^2-z)i+(y^2+z)j+xk z=x^2 x^2+y^2

In Exercises 37 and38, use a computer algebra system to evaluate the line integral. \(\int_{C} x y d x+\left(x^{2}+y^{2}\right) d y\) \(C: y=x^{2}\) from (0, 0) to (2, 4) and y = 2x from (2, 4) to (0, 0) Text Transcription: int_C xydx+(x^2+y^2)dy C:y=x^2

In Exercises 37 and38, use a computer algebra system to evaluate the line integral. \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) F(x, y) = (2x - y)i + (2y - x)j C: r(t) = (2 cos t + 2t sin t)i + (2 sin t - 2t cos t)j, \(0 \leq t \leq \pi\) Text Transcription: int_C F cdot dr 0 <= t <= pi

Work Find the work done by the force field \(\mathbf{F}=x \mathbf{i}-\sqrt{y} \mathbf{j}\) along the path \(y=x^{3 / 2}\) from (0, 0) to (4, 8). Text Transcription: F=xi-sqrt y j y=x^3/2

Work A 20-ton aircraft climbs 2000 feet while making a \(90^{\circ}\) turn in a circular arc of radius 10 miles. Find the work done by the engines. Text Transcription: 90^o

In Exercises 41 and 42, evaluate the integral using the Fundamental Theorem of Line Integrals. \(\int_{C} 2 x y z d x+x^{2} z d y+x^{2} y d z\) C: smooth curve from (0, 0, 0) to (1, 3, 2) Text Transcription: int_C 2xyz dx+x^2z dy+x^2y dz

In Exercises 41 and 42, evaluate the integral using the Fundamental Theorem of Line Integrals. \(\int_{C} y d x+x d y+\frac{1}{z} d z\) C: smooth curve from (0, 0, 1) to (4, 4, 4) Text Transcription: int_C y dx+x dy+1/z dz

Evaluate the line integral \(\int_{C} y^{2} d x+2 x y d y\). (a) C: r(t) = (1 + 3t)i + (1 + t)j , \(0 \leq t \leq 1\) (b) \(C: \mathbf{r}(t)=t \mathbf{i}+\sqrt{t} \mathbf{j}, \quad 1 \leq t \leq 4\) (c) Use the Fundamental Theorem of Line Integrals, where C is a smooth curve from (1, 1) to (4, 2). Text Transcription: int_C y^2 dx+2xy dy 0<=t<=1 C:r(t)=ti+sqrt t j, 1<=t<=4

Area and Centroid Consider the region bounded by the x-axis and one arch of the cycloid with parametric equations \(x=a(\theta-\sin \theta)\) and \(y=a(1-\cos \theta)\). Use line integrals to find (a) the area of the region and (b) the centroid of the region. Text Transcription: x=a(theta-sin theta) y=a(1-cos theta)

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C} y d x+2 x d y\) C: boundary of the square with vertices (0, 0), (0, 1), (1, 0), (1, 1) Text Transcription: int_C y dx+2x dy

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C} x y d x+\left(x^{2}+y^{2}\right) d y\) C: boundary of the square with vertices (0, 0), (0, 2), (2, 0), (2, 2) Text Transcription: int_C xy dx+(x^2+y^2)dy

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C} x y^{2} d x+x^{2} y d y\) C: x = 4 cos t, y = 4 sin t Text Transcription: int_C xy^2 dx+x^2y dy

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C}\left(x^{2}-y^{2}\right) d x+2 x y d y\) \(C: x^{2}+y^{2}=a^{2}\) Text Transcription: int_C (x^2-y^2)dx+2xy dy C:x^2+y^2=a^2

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C} x y d x+x^{2} d y\) C: boundary of the region between the graphs of \(y=x^{2}\) and y = 1 Text Transcription: int_C xy dx+x^2 dy y=x^2

In Exercises 45-50, use Green's Theorem to evaluate the line integral. \(\int_{C} y^{2} d x+x^{4 / 3} d y\) \(C: x^{2 / 3}+y^{2 / 3}=1\) Text Transcription: int_C y^2 dx+x^4/3 dy C: x^2/3+y^2/3=1

In Exercises 51 and 52, use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = sec u cos vi + (1 + 2 tan u)sin vj + 2uk \(0 \leq v \leq 2 \pi\) Text Transcription: 0 <= v <=2 pi

In Exercises 51 and 52, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=e^{-u / 4} \cos v \mathbf{i}+e^{-u / 4} \sin v \mathbf{j}+\frac{u}{6} \mathbf{k}\) \(0 \leq u \leq 4, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u,v)=e^-u/4 cos vi+e^-u/4 sin vj+u/6 k 0 <= u <= 4, 0 <= v <= 2 pi

Investigation Consider the surface represented by the vector valued function r(u, v) = 3 cos v cos ui + 3 cos v sin uj + sin vk Use a computer algebra system to do the following. (a) Graph the surface for \(0 \leq u \leq 2 \pi\) and \(-\frac{\pi}{2} \leq v \leq \frac{\pi}{2}\). (b) Graph the surface for \(0 \leq u \leq 2 \pi\) and \(\frac{\pi}{4} \leq v \leq \frac{\pi}{2}\). (c) Graph the surface for \(0 \leq u \leq \frac{\pi}{4}\) and \(0 \leq v \leq \frac{\pi}{2}\). (d) Graph and identify the space curve for \(0 \leq u \leq 2 \pi\) and \(v=\frac{\pi}{4}\) (e) Approximate the area of the surface graphed in part (b). (f) Approximate the area of the surface graphed in part (c). Text Transcription: 0<=u<=2 pi -pi/2<=v<=pi/2 pi/2<=v<=pi/2 0<=u<=pi/4 0<=v<=pi/2 v=pi/4

Evaluate the surface integral \(\int_{S} \int z d S\) over the surface S: r(u, v) = (u+v)i + (u-v)j + sin vk where \(0 \leq u \leq 2\) and \(0 \leq v \leq \pi\). Text Transcription: int_S int z dS 0 <= u <= 2 0 <= v <= pi

Use a computer algebra system to graph the surface S and approximate the surface integral \(\int S \int(x+y) d S\) where S is the surface S: r(u, v) = u cos vi + u sin vj + (u - 1)(2 - u)k over \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\). Text Transcription: int S int(x+y) dS 0 <= u <= 2 0 <= v <= 2 pi

Mass A cone-shaped surface lamina S is given by \(z=a\left(a-\sqrt{x^{2}+y^{2}}\right), \quad 0 \leq z \leq a^{2}\) At each point on S, the density is proportional to the distance between the point and the z-axis. (a) Sketch the cone-shaped surface. (b) Find the mass m of the lamina. Text Transcription: z=a(a-sqrt x^2+y^2), 0<=z<=a^2

In Exercises 57 and 58, verify the Divergence Theorem by evaluating \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+x y \mathbf{j}+z \mathbf{k}\) Q: solid region bounded by the coordinate planes and the plane 2x + 3y + 4z = 12 Text Transcription: int_S int F cdot N dS F(x,y,z)=x^2i+xyj+zk

In Exercises 57 and 58, verify the Divergence Theorem by evaluating \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) as a surface integral and as a triple integral. F(x,y,z) = xi + yj + zk Q: solid region bounded by the coordinate planes and the plane 2x + 3y + 4z = 12 Text Transcription: int_S int F cdot N dS

In Exercises 59 and 60, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. F(x,y,z) = (cos y + y cos x)i + (sin x - x sin y)j + xyzk S: portion of \(z=y^{2}\) over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), (0, a) N is the upward unit normal vector to the surface. Text Transcription: int_C F cdot d r z=y^2

In Exercises 59 and 60, verify Stokes's Theorem by evaluating \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) as a line integral and as a double integral. \(\mathbf{F}(x, y, z)=(x-z) \mathbf{i}+(y-z) \mathbf{j}+x^{2} \mathbf{k}\) S: first-octant portion of the plane 3x + y + 2z = 12 Text Transcription: int_C F cdot d r F(x,y,z)=(x-z)i+(y-z)j+x^2k

Prove that it is not possible for a vector field with twice differentiable components to have a curl of xi + yj + zk.