Given a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3,\)
compute its divergence and its curl.
Compute the value of a line integral
over a piecewise-smooth curve in a vector field.
If the curve is closed, compute its value either directly
or as an integral over its interior
via either Green’s Theorem or Stokes’ Theorem.
Compute the value of a surface integral
over a piecewise smooth surface in a vector field.
If the surface is closed, compute its value either directly
or as an integral over its interior
via either Green’s Theorem or the Divergence Theorem.
Exercises
Evaluate \(\int_C x^4\,\mathrm{d}x + xy\,\mathrm{d}y,\)
where \(C\) is the triangular curve consisting of
the line segments from \((0,0)\) to \((1,0)\)
and from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((0,0).\)
Evaluate \(\oint \Big(3y-\mathrm{e}^{\sin(x)}\Big)\,\mathrm{d}x + \Big(7x+\sqrt{y^4+1}\Big)\,\mathrm{d}y,\)
where \(C\) is the circle \(x^2+y^2=9.\)
Calculate the area enclosed by the ellipse
\(\frac{1}{a^2}x^2+\frac{1}{b^2}y^2=1.\)
Evaluate \(\oint y^2\,\mathrm{d}x + 3xy\,\mathrm{d}y,\)
where \(C\) is the boundary of the semiannular region \(R\)
in the upper half-plane between the circles
\(x^2+y^2=1\) and \(x^2+y^2=4.\)
show that \(\int_C \bm{F}\cdot\mathrm{d}\bm{r} = 2\pi\)
for every positively oriented simple closed path
that encloses the origin.
For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\)
compute the curl of \(\bm{F}.\)
Show that the vector field \(\bm{F}\) defined as
\(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\)
is not conservative.
Show that the vector field \(\bm{F}\) defined as
\(\bm{F}(x,y,z) = y^2 z^3\mathbf{i} + 2xyz^3 \mathbf{j} +3xy^2z^2\mathbf{k}\)
is a conservative vector field,
and find an example of a function \(f\)
such that \(\bm{F} = \nabla f.\)
For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\)
compute the divergence of \(\bm{F}.\)
Show that the vector field \(\bm{F}\) defined as
\(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\)
cannot be written as the curl of another vector field.
I.e. there is no vector field \(\bm{G}\)
such that \(\bm{F} = \operatorname{curl}\bm{G}.\)
Evaluate \(\int_C \bm{F}\cdot\mathrm{d}\bm{r},\)
where \(\bm{F}(x,y,z) = -y^2\mathbf{i} +x\mathbf{j}+z^2\mathbf{k}\)
and \(C\) is the curve of intersection of the plane
\(y+z=2\) and the cylinder \(x^2+y^2=1.\)
(Orient \(C\) to be counterclockwise when viewed from above.)
Use Stokes’ Theorem to compute the integral
\(\iint_S \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{S},\)
where \(\bm{F}(x,y,z) = xz\mathbf{i}+yz\mathbf{j}+xy\mathbf{k}\)
and \(S\) is the part of the sphere \(x^2+y^2+z^2=4\)
that lies inside the cylinder \(x^2+y^2=1\)
and above the \(xy\)-plane.
Calculate the flux of the vector field
\(\bm{F}(x,y,z) = z\mathbf{i}+y\mathbf{j}+x\mathbf{k}\)
across the unit sphere \(x^2+y^2+z^2=1.\)
Evaluate \(\iint_S \bm{F}\cdot\mathrm{d}\bm{S},\) where
\[ \bm{F}(x,y,z) = xy\mathbf{i} + \Big(y^2+\mathrm{e}^{xz^2}\Big)\mathbf{j} + \sin(xy)\mathbf{k} \]
and \(S\) is the surface of the region \(E\) bounded by
the parabolic cylinder \(z=1-x^2\)
and the planes \(z=0\) and \(y=0\) and \(y+z=2.\)
Problems & Challenges
Schey
Let \(\mathbf{i}\) and \(\mathbf{j}\) and \(\mathbf{k}\)
be the unit basis vectors of the standard rectangular coordinate system on \(\mathbf{R}^3,\)
and let \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\) and \(\mathbf{e}_z\)
be the unit basis vectors of the standard cylindrical coordinate system on \(\mathbf{R}^3.\)
Show that
\[\begin{align*}
\mathbf{i} &= \mathbf{e}_r\cos(\theta) - \mathbf{e}_\theta\sin(\theta)
\\\mathbf{j} &= \mathbf{e}_r\sin(\theta) + \mathbf{e}_\theta\cos(\theta)
\\\mathbf{k} &= \mathbf{e}_z\,.
\end{align*}\]
For a vector field \(\bm{F}(r,\theta,z) = P\mathbf{e}_r + Q\mathbf{e}_\theta + R\mathbf{e}_z,\)
show that \(\operatorname{div}\bm{F}\) is given by
\[\operatorname{div}\bm{F} =
\frac{1}{r}\frac{\partial}{\partial r}\big(rP\big)
+ \frac{1}{r}\frac{\partial Q}{\partial \theta}
+ \frac{1}{r}\frac{\partial R}{\partial z}\,.
\]
Determine an analogous formula
for \(\operatorname{curl}\bm{F}\) in cylindrical coordinates.
Schey
Let \(\mathbf{i}\) and \(\mathbf{j}\) and \(\mathbf{k}\)
be the unit basis vectors of the standard rectangular coordinate system on \(\mathbf{R}^3,\)
and let \(\mathbf{e}_\rho\) and \(\mathbf{e}_\theta\) and \(\mathbf{e}_\varphi\)
be the unit basis vectors of the standard spherical coordinate system on \(\mathbf{R}^3.\)
For a vector field
\(\bm{F}(\rho,\theta,\varphi) = P\mathbf{e}_\rho + Q\mathbf{e}_\theta + R\mathbf{e}_\varphi,\)
show that \(\operatorname{div}\bm{F}\) is given by
\[
\operatorname{div}\bm{F} =
\frac{1}{\rho^2}\frac{\partial}{\partial \rho}\big(r^2P\big)
+ \frac{1}{\rho\sin(\theta)}\frac{\partial}{\partial \theta}\big(\sin(\theta)Q\big)
+ \frac{1}{\rho\sin(\theta)}\frac{\partial R}{\partial \varphi}
\]
Determine an analogous formula
for \(\operatorname{curl}\bm{F}\) in spherical coordinates.
Schey
For an arbitrary vector \(\bm{v}\) and \(\bm{r} = \langle x,y,z\rangle,\)
show that \[ \operatorname{curl}\Bigl(\frac{\bm{v}\times\bm{r}}{2}\Bigr) = \bm{v}\,. \]
Schey
Verify Stokes’ Theorem for these surfaces \(S\)
and vector fields \(\bm{F}.\)
The surface \(S\) consisting of the five faces of the unit cube in the first quadrant
that don’t lie in the \(xz\) plane,
and \(\bm{F} = \langle z^2, -y^2, 0\rangle.\)
The surface that is the eighth of the unit sphere
centered at the origin
that lies entirely inside the first quadrant,
and \(\bm{F} = \langle y,z,x \rangle.\)
Stewart
Prove the following identity:
\[
\nabla\bigl(\bm{F}\cdot\bm{G}\bigr)
= \bigl(\bm{F}\cdot\nabla\bigr) \bm{G}
+ \bigl(\bm{G}\cdot\nabla\bigr) \bm{F}
+ \bm{F} \times \operatorname{curl}\bm{G}
+ \bm{G} \times \operatorname{curl}\bm{F}
\]
Schey
Recalling that \(\operatorname{div} = (\nabla\cdot)\)
and that \(\operatorname{grad} = (\nabla)\)
and that \(\operatorname{curl} = (\nabla\times)\)
and that \(\nabla^2\) is the Laplace operator,
verify the following identities concerning
arbitrary differentiable scalar functions \(f\) and \(g\)
and arbitrary differentiable vector fields \(\bm{F}\) and \(\bm{G}.\)
Stewart
Find the positively oriented simple closed curve \(C\)
for which the value of the line integral is a maximum.
\[ \oint\limits_C \bigl(y^3-y\bigr)\,\mathrm{d}x - \bigl(2x^3\bigr)\,\mathrm{d}y \]