Chapter 14

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=\langle-y, x\rangle$$

Find the curl and divergence of the given vector field. $$x^{2} \mathbf{i}-3 x y \mathbf{j}$$

Verify the Divergence Theorem by computing both integrals. $$\begin{aligned} &\mathbf{F}=\left\langle 2 x z, y^{2},-x z\right\rangle, Q \text { is the cube } 0 \leq x \leq 1,0 \leq y \leq 1\\\ &0 \leq z \leq 1 \end{aligned}$$

\(S\) is the portion of \(z=4-x^{2}-y^{2}\) above the \(x y\) -plane, \(\mathbf{F}=\left\langle z x, 2 y, z^{3}\right\rangle\)

Find a parametric representation of the surface. $$z=3 x+4 y$$

Evaluate the indicated line integral (a) directly and (b) using Green's Theorem. \(\oint_{C}\left(x^{2}-y\right) d x+y^{2} d y,\) where \(C\) is the circle \(x^{2}+y^{2}=1\) oriented counterclockwise

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle 2 x y-1, x^{2}\right\rangle$$

Evaluate the line integral. \(\int_{C} 2 x d s,\) where \(C\) is the line segment from (1,2) to (3,5)

Find the curl and divergence of the given vector field. $$y^{2} \mathbf{i}+4 x^{2} y \mathbf{j}$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=\frac{\langle-y, x\rangle}{\sqrt{x^{2}+y^{2}}}$$

Verify the Divergence Theorem by computing both integrals. $$\mathbf{F}=\langle x, y, z\rangle, Q \text { is the ball } x^{2}+y^{2}+z^{2} \leq 1$$

Verify Stokes' Theorem by computing both integrals. \(S\) is the portion of \(z=1-x^{2}-y^{2}\) above the \(x y\) -plane, \(\mathbf{F}=\left\langle x^{2} z, x y, x z^{2}\right\rangle\)

Evaluate the indicated line integral (a) directly and (b) using Green's Theorem. \(\oint_{C}\left(y^{2}+x\right) d x+(3 x+2 x y) d y,\) where \(C\) is the circle \(x^{2}+y^{2}=4\) oriented counterclockwise

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle 3 x^{2} y^{2}, 2 x^{3} y-y\right\rangle$$

Evaluate the line integral. \(\int_{C}(x-y) d s,\) where \(C\) is the line segment from (1,0) to (3,1)

Find the curl and divergence of the given vector field. $$2 x z \mathbf{i}-3 y \mathbf{k}$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=\left\langle 0, x^{2}\right\rangle$$

Verify the Divergence Theorem by computing both integrals. $$\mathbf{F}=\left\langle x z, z y, 2 z^{2}\right\rangle, Q \text { is bounded by } z=1-x^{2}-y^{2} \text { and } z=0$$

Verify Stokes' Theorem by computing both integrals. \(S\) is the portion of \(z=\sqrt{4-x^{2}-y^{2}}\) above the \(x y\) -plane, \(\mathbf{F}=\left\langle 2 x-y, y z^{2}, y^{2} z\right\rangle\)

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\langle y z, x z, x y\rangle,\) find the total charge in the hemisphere \(z=\sqrt{R^{2}-x^{2}-y^{2}}\).

Find a parametric representation of the surface. $$x^{2}+y^{2}-z^{2}=1$$

Evaluate the indicated line integral (a) directly and (b) using Green's Theorem. \(\oint_{C} x^{2} d x-x^{3} d y,\) where \(C\) is the square from (0,0) to (0,2) to (2,2) to (2,0) to (0,0)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle\frac{1}{y}-2 x, y-\frac{x}{y^{2}}\right\rangle$$

Evaluate the line integral. \(\int_{C}(3 x+y) d s,\) where \(C\) is the line segment from (5,2) to (1,1)

Find the curl and divergence of the given vector field. $$x^{2} \mathbf{i}-3 x y \mathbf{j}+x \mathbf{k}$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=\langle 2 x, 0\rangle$$

Verify the Divergence Theorem by computing both integrals. \(\mathbf{F}=\left\langle x^{2}, 2 y,-x^{2}\right\rangle, \quad Q \quad\) is the tetrahedron bounded by \(x+2 y+z=4\) and the coordinate planes

Verify Stokes' Theorem by computing both integrals. \(S\) is the portion of \(z=\sqrt{1-x^{2}-y^{2}}\) above the \(x y\) -plane, \(\mathbf{F}=\left\langle 2 x, z^{2}-x, x z^{2}\right\rangle\)

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\left\langle 2 x y, y^{2}, 5 x\right\rangle,\) find the total charge in the hemisphere \(z=\sqrt{R^{2}-x^{2}-y^{2}}\).

Evaluate the indicated line integral (a) directly and (b) using Green's Theorem. \(\oint_{C}\left(y^{2}-2 x\right) d x+x^{2} d y,\) where \(C\) is the square from (0,0) to (1,0) to (1,1) to (0,1) to (0,0)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\langle\sin y-x, x \cos y\rangle$$

Evaluate the line integral. \(\int_{C} 2 x y d s,\) where \(C\) is the line segment from (1,2) to (-1,0)

Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of the tetrahedron bounded by \(x+y+2 z=2\) and the coordinate planes with \(z>0,\) n upward, \(\mathbf{F}=\left\langle z y^{4}-y^{2}, y-x^{3}, z^{2}\right\rangle\)

Find the curl and divergence of the given vector field. $$\left\langle x y, y z, x^{2}\right\rangle$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\) \(Q\) is bounded by \(x+y+2 z=2\) (first octant) and the coordinate planes, \(\mathbf{F}=\left\langle 2 x-y^{2}, 4 x z-2 y, x y^{3}\right\rangle\)

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=(4 x-y, 2 y+z, 3 x y),\) find the total charge in the hemisphere \(z=\sqrt{R^{2}-x^{2}-y^{2}}\).

Find a parametric representation of the surface. The portion of \(x^{2}+y^{2}=4\) from \(z=0\) to \(z=2\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} x e^{2 x} d x-3 x^{2} y d y,\) where \(C\) is the rectangle from (0,0) to (3,0) to (3,2) to (0,2) to (0,0)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle e^{x y}-1, x e^{x y}\right\rangle$$

Evaluate the line integral. \(\int_{C} 2 x d x,\) where \(C\) is the line segment from (0,2) to (2,6)

Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of the tetrahedron bounded by \(x+y+4 z=8\) and the coordinate planes with \(z>0,\) \(\mathbf{n}\) upward, \(\mathbf{F}=\left\langle y^{2}, y+2 x, z^{2}\right\rangle\)

Find the curl and divergence of the given vector field. $$\left\langle x e^{z}, y z^{2}, x+y\right\rangle$$

Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=-\mathbf{i}+y^{2} \mathbf{j}$$

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q\) is bounded by \(4 x+2 y-z=4(z \leq 0)\) and the coordinate planes, \(\mathbf{F}=\left\langle x^{2}-y^{2} z, x \sin z, 4 y^{2}\right\rangle\)

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\left\langle 2 x z^{2}, 2 y x^{2}, 2 z y^{2}\right\rangle,\) find the total charge in the hemisphere \(z=\sqrt{R^{2}-x^{2}-y^{2}}\).

Find a parametric representation of the surface. The portion of \(y^{2}+z^{2}=9\) from \(x=-1\) to \(x=1\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} y e^{2 x} d x+x^{2} y^{2} d y,\) where \(C\) is the rectangle from (-2,0) to (3,0) to (3,2) to (-2,2) to (-2,0)

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y)=\left\langle e^{y}-2 x, x e^{y}-x^{2} y\right\rangle$$

Evaluate the line integral. \(\int_{C} 3 y^{2} d y,\) where \(C\) is the line segment from (2,0) to (1,3)

Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of \(z=1-x^{2}-y^{2}\) above the \(x y\) -plane with \(\mathbf{n}\) upward, \(\mathbf{F}=\left\langle z x^{2}, z e^{x y^{2}}-x, x \ln y^{2}\right\rangle\)