Question

True or False The curl of a vector field \(\mathbf{F}\) is a scalar.

Question

If the vector field \(\,\mathbf{F}=( e^{-x^{2}/2}-yz) \mathbf{i}+( e^{-y^{2}/2}+xz+2x) \mathbf{j}+( e^{-z^{2}/2}+5) \mathbf{k}\), then \(\rm{curl}\, \mathbf{F}\)=________.

Question

Multiple Choice A vector field \(\mathbf{F}\) is conservative if and only if \(\rm{curl}\, \mathbf{F}=\) [(a) \(\mathbf{F}\) , (b) div \(\mathbf{F}\), (c) \(\mathbf{0}\), (d) \(\mathbf{n}\)].

Question

True or False An interpretation of the curl of \( \mathbf{F}\) is circulation per unit area of a fluid at a given point on a surface.

Question

Suppose \(\mathbf{F}\) is the velocity vector of a fluid rotating about a fixed axis and that \({\bf\omega }\) is a constant angular velocity. Then \(\rm{curl}\, \mathbf{F}=\)________.

Question

True or False If \(\mathbf{F}\) is a conservative vector field, then the work done by \(\mathbf{F}\) in moving an object of mass \(m\) from point \(A\) to point \(B\) depends on the path taken from \(A\) to \(B\).

Question

\(\mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}\)

Question

\(\mathbf{F}(x,y,z)=y\mathbf{i}+x\mathbf{j}\)

Question

\(\mathbf{F}(x,y,z)=xyz\mathbf{i}+xz\mathbf{j}+z\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=4x \mathbf{i}-y\mathbf{j}-2z\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=3xyz^{2}\mathbf{i}+y^{2}\sin z\, \mathbf{j}+xe^{2z}\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=yz\mathbf{i}+z^{2}x\mathbf{j}+yz\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=\dfrac{x\mathbf{i}}{x^{2}+y^{2}+z^{2}}+ \dfrac{y\mathbf{j}}{x^{2}+y^{2}+z^{2}}+\dfrac{\mathbf{k}}{x^{2}+y^{2}+z^{2}}\)

Question

\(\mathbf{F}(x,y,z)=e^{x}\mathbf{i}+x^{2}y\mathbf{j}+e^{z}\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=\cos x\mathbf{i}+\sin y\mathbf{j}+e^{xz}\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=\sin (xy) \,\mathbf{i}+\cos ( xy^{2}) \,\mathbf{j}+x\,\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=(x+y)\,\mathbf{i}+(y+z)\,\mathbf{j}+(z+x)\,\mathbf{k}\)

Question

\(\mathbf{F}(x,y,z)=(y+z)\,\mathbf{i}+(z+x)\,\mathbf{j}+(z+y+x)\,\mathbf{k}\)

Question

\(\mathbf{F}=(z-y)\mathbf{i}+(z+x)\mathbf{j}-(x+y) \mathbf{k}\); \(S\) is the portion of the paraboloid \(z=1-x^{2}-y^{2},\) \(z\geq 0 \).

Question

\(\mathbf{F}=y\mathbf{i}+z\mathbf{j}+x\mathbf{k}\); \(S\) is the portion of the paraboloid \(z=1-x^{2}-y^{2}\), \(z\geq 0\).

Question

\(\mathbf{F}=y\mathbf{i}-x\mathbf{j}\); \(S\) is the hemisphere \(z=\sqrt{1-x^{2}-y^{2}} \).

Question

\(\mathbf{F}=z\mathbf{i}+x\mathbf{j}+y\mathbf{k}\); \(S\) is the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\).

Question

\(\mathbf{F}=y^{2}\mathbf{i}+x\mathbf{j}-xz\mathbf{k}\); \(S\) is the surface \(z=1-x^{2}-y^{2}\), \(z\geq 0\).

Question

\(\mathbf{F}(x,y,z)=4x\mathbf{i}-y\mathbf{j}+2z\mathbf{k}\); \(S\) is the surface \(z=1+x^{2}+y^{2}\), \(z\leq 5\).

Question

Use Stokes' Theorem to find the surface integral \(\iint\limits_{\kern-3ptS}\hbox{curl}\,\mathbf{F} \,{\bf\cdot}\, \mathbf{n}\,dS,\) where \(\mathbf{F}=\mathbf{F}( x,y,z) =y \mathbf{i}+x\mathbf{j}+x^{2}\mathbf{k}\) and \(S\) is the surface enclosed by the paraboloid \(z=9-\) \(x^{2}-y^{2},\) \(z\geq 0\).

Question

Use Stokes' Theorem to find the surface integral \(\iint\limits_{\kern-3ptS}\hbox{curl}\,\mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS,\) where \(\mathbf{F}=\mathbf{F}( x,y,z) =4z\mathbf{i }+3x\mathbf{j}+3y\mathbf{k}\) and \(S\) is the surface enclosed by the paraboloid \(z=10-x^{2}-y^{2},\) \(z\geq 4\).

Question

\(\oint_{C}[(y+z)\,dx+(z+x)\,dy+(x+y)\,dz];\) \(C\) is the curve of intersection of \(x^{2}+y^{2}+z^{2}=1\), and \(x+y+z=0\).

Question

\(\oint_{C}[(y-z)\,dx+(z-x)\,dy+(x-y)\,dz];\) \(C\) is the curve of intersection of \( x^{2}+y^{2}=1\), and \(x+z=1\).

Question

\(\oint_{C}[x\,dx+(x+y)\,dy+(x+y+z)\,dz];\) \(C\) is the curve \( x=2\,\cos t\), \(y=2\,\sin t\), \(z=2\), \(0\leq t\leq 2\pi \).

Question

\(\oint_{C}(y^{2}\,dx+z^{2}\,dy+x^{2}\,dz);\) \(C\) is the triangle with vertices \((1,0,0),(0,1,0)\), and \((0,0,1)\).

Question

\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}\)

Question

\(\mathbf{F}=y\mathbf{i}+x\mathbf{j}\)

Question

\(\mathbf{F}=xy\mathbf{i}+yz\mathbf{j}+zx\mathbf{k}\)

Question

\(\mathbf{F}=yz\mathbf{i}+zx\mathbf{j}+xy\mathbf{k}\)

Question

Find the value of the constant \(c\) so that \(\mathbf{F}=xy \mathbf{i}+cx^{2}\mathbf{j}\) in space is a conservative vector field.

Question

Find the value of the constant \(c\) so that \(\mathbf{F}=\dfrac{ z}{y}\mathbf{i}+c\dfrac{xz}{y^{2}}\mathbf{j}+\dfrac{x}{y}\mathbf{k}\), \( y\neq 0\), is a conservative vector field.

Question

Show that \(\rm{curl}\,(\mathbf{F}+\mathbf{G})=\rm{curl}\, \mathbf{F}+\rm{curl}\,\mathbf{G}\).

Question

Show that \(\rm{curl}\,\left( c\mathbf{F}\right) =c(\rm{curl}\, \mathbf{F})\), where \(c\) is a constant.

Question

If \(\ \mathbf{F}(x,y,z)=z\mathbf{i}+x\mathbf{j}+y\mathbf{k}\), find \(\iint\limits_{\kern-3ptS}\mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS\), where \(S\) is the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\).

Question

Rework Problem 39, where \(S\) is the circular region \( x^{2}+y^{2}\leq 1\), \(z=0\).

Question

Show that \(\mathbf{F}=y\mathbf{i}-x\mathbf{j}+z\mathbf{k}\) is not a conservative vector field. Nevertheless, there are certain paths \(C\) for which \(\oint_{C}\mathbf{F}\,{\bf\cdot}\, d\mathbf{r}=0\). Find one.

Question

Show that div \(\left( \rm{curl}\, \mathbf{F}\right) = \mathbf{0}\), where \(\mathbf{F}=P( x,y,z) \mathbf{i}+ Q( x,y,z) \mathbf{j}+R( x,y,z) \mathbf{k}\) and \(P\), \(Q\), and \( R \) are twice differentiable and the partial derivatives are continuous.

Question

Suppose \(\mathbf{F}(x,y,z)=P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{ j}+R(x,y,z)\mathbf{k}\) is a vector field with continuous and differentiable components in a simply connected region of space.

1. Use Stokes' Theorem to show that \(\mathbf{F}\) is a gradient field if \(Q_{x}=P_{y}\), \(R_{y}=Q_{z}\), and \(P_{z}=R_{x}\).
2. Why does it follow that \(\int_{C}\mathbf{F}\,{\bf\cdot}\, d\mathbf{r}\) is independent of the path when these conditions hold?

Question

Use Problem 43 to show that \(\int_{C}[(yz-y-z)\,dx+ (xz-x-z) \,dy+(xy-x-y)\,dz]\) is independent of the path.

Question

Work An object is moved from the origin to the point \( (a,b,c)\) in the field of force \(\mathbf{F}=(x+y)\mathbf{i}+(x-z)\mathbf{j} +(z-y)\mathbf{k}\). Show that the work done depends on only \(a,b\), and \(C\), and find this value.

Question

Show that no twice differentiable vector function exists whose curl is \(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\).

Question

Let \(\mathbf{F}(x,y,z)\) be a vector field with continuous and differentiable components, and let \(S\) be a sphere with outer unit normal \( \mathbf{n}\). Use Stokes' Theorem to show that \(\iint\limits_{\kern-3ptS}{\rm{curl}\, \mathbf{F}\,{\bf\cdot}\, \mathbf{n}}\,dS=0\).

Question

Assume that the hypotheses of the Divergence Theorem hold for \( S\) and \(E.\) Show that for any vector field \(\mathbf{F}\) with continuous first-order partial derivatives in a closed, bounded solid \(E\) with orientable boundary \(S\) with outer unit normal \(\mathbf{n}\), \(\ \iint\limits_{\kern-3ptS}{\rm{curl}\, \mathbf{F}\,{\bf\cdot}\, \mathbf{n}}\,dS=0.\)

Question

Let \(C\) be a smooth, simple, closed curve lying on an orientable surface \(S,\) and let \(f\) and \(g\) have continuous partial derivatives on \(S\). Show that \(\int_{C}(f{\bf\nabla }g)\,{\bf\cdot}\, d\mathbf{r} =\iint\limits_{\kern-3ptS}(\bf\nabla\ f\times {\bf\nabla }g)\,{\bf\cdot}\, \mathbf{n} \,dS\). Assume that the portion of \(S\) bounded by \(C\) is smooth and simply connected.