Given a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3,\)
determine whether or not it’s conservative.
If it’s conservative, determine its potential function.
Exercises
Consider the vector field
\(\bm{F}\colon\mathbf{R}^2\to\mathbf{R}^2\)
defined as
\(\bm{F}(x,y) = -y\mathbf{i}+x\mathbf{j}.\)
Sketch some of the vectors of this vector field
until you have a good idea of what it looks like.
Sketch the vector field
\(\bm{F}\colon\mathbf{R}^3\to\mathbf{R}^3\)
defined as \(\bm{F}(x,y,z) = z\mathbf{k}.\)
Sketch the gradient field and a contour plot
of the function \(f\) defined as \(f(x,y) = x^2-y^3\)
and notice how they’re related.
Evaluate the line integral \(\int_C 2+x^2y\,\mathrm{d}s\)
where \(C\) is the upper half of the unit circle \(x^2+y^2=1.\)
Evaluate \(\int_C y^2\,\mathrm{d}x + x\,\mathrm{d}y\)
along two different paths \(C:\)
first the line segment from \((-5,-3)\) to \((0,2),\)
then the arc of the parabola \(x=4-y^2\)
from \((-5,-3)\) to \((0,2).\)
Evaluate \(\int_C y\sin(z)\,\mathrm{d}s\)
where \(C\) is the circular helix defined parametrically
by the equations \(x=\cos(t)\) and \(y=\sin(t)\) and \(z=t\)
for \(0\leq t \leq 2\pi.\)
Evaluate the line integral
\(\int_C y\,\mathrm{d}x + z\,\mathrm{d}y + x\,\mathrm{d}z \)
where \(C\) consists of the line segment
\(C_1\) from \((2,0,0)\) to \((3,4,5),\)
followed by the vertical line segment
\(C_2\) from \((3,4,5)\) to \((3,4,0).\)
Calculate the work done by the force field
\(\bm{F}(x,y) = x^2\mathbf{i}-xy\mathbf{j}\)
in moving a particle along the quarter-circle
\(\bm{r}(t) = \cos(t)\mathbf{i}+\sin(t)\mathbf{j}\)
for \(0 \leq t \leq \pi/2.\)
Evaluate \(\int_C \bm{F}\cdot\mathrm{d}\bm{r}\)
where \(\bm{F}(x,y,z) = xy\mathbf{i} + yz\mathbf{j} + xz\mathbf{k}\)
and \(C\) is the twisted cubic give by the equations
\(\langle x,y,z \rangle = \langle t,t^2,t^3 \rangle\)
for \(0\leq t \leq 1.\)
Calculate the work done by the gravitational field
For \(\bm{F}(x,y) = (3+2xy)\mathbf{i} + (x^2-3y^2)\mathbf{j},\)
find a function \(f\colon \mathbf{R}^2\to\mathbf{R}\)
such that \(\bm{F} = \nabla f.\)
Evaluate the line integral \(\int_C \bm{F}\cdot\mathrm{d}\bm{r}\)
where \(\bm{F}(x,y) = (3+2xy)\mathbf{i} + (x^2-3y^2)\mathbf{j}\)
and \(C\) is defined as the curve
\(\bm{r}(t) = \mathrm{e}^t\sin(t)\mathbf{i} + \mathrm{e}^t\cos(t)\mathbf{j}\)
for \(0 \leq t \leq \pi.\)
For \(\bm{F}(x,y,z) = y^2\mathbf{i} + (2xy+\mathrm{e}^{3z})\mathbf{j} + 3y\mathrm{e}^{3z}\mathbf{k}\)
find a function \(f\colon \mathbf{R}^3\to\mathbf{R}\)
such that \(\bm{F} = \nabla f.\)
Compute the surface integral \(\iint_S x^2 \,\mathrm{d}S\)
where \(S\) is the unit sphere \(x^2+y^2+z^2=1.\)
Evaluate \(\iint_S y \,\mathrm{d}S,\)
where \(S\) is the surface \(z=x+y^2\)
for \(0\leq x \leq 1\) and \(0\leq y \leq 2.\)
Evaluate \(\iint_S z \,\mathrm{d}S,\) where \(S\) is the surface
whose sides \(S_1\) are given by the cylinder \(x^2+y^2=1,\)
whose bottom \(S_2\) is the disk \(x^2+y^2 \leq 1\) in the plane \(z=0,\)
and whose top \(S_3\) is the part of the plane \(z=1+x\) that lines above \(S_2.\)
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 \(\iint_S \bm{F}\cdot\mathrm{d}S,\)
where \(\bm{F}(x,y,z) = y\mathbf{i}+x\mathbf{j}+z\mathbf{k}\)
and \(S\) is the boundary of the solid enclosed by
the paraboloid \(z=1-x^2-y^2\) and the plane \(z=0.\)
The temperature \(u\) in a metal ball is proportional
to the square of the distance from the center of the ball.
Calculate the rate of heat flow across a sphere of radius \(r\)
with center at the center of the ball.
Problems & Challenges
Schey
Instead of using arrows to represent vector functions,
we can use families of curves called field lines.
In two dimensions, a curve \(\bigl(x(t), y(t)\bigr)\)
is a field line of the vector field \(\bm{F}\)
if at each point \((x_0, y_0)\) on the curve,
\(\bm{F}(x_0,y_0)\) is tangent to the curve.
Realize that the field lines \(\bigl(x(t), y(t)\bigr)\)
for a vector field \(\bm{F} = P\mathbf{i} + Q\mathbf{j}\)
are solutions to the differential equation
\[ \frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t} = \frac{Q}{P}\,. \]
Determine the family of field lines for each of these vector fields.
Schey
Sometimes the value of a surface integral can be reasoned out
without working through the usual arduous procedure
simply by remembering that the surface integral calculates the flux,
the “sum of flow”, across the surface.
For each of the following vector fields \(\bm{F}\) and surfaces \(S,\)
reason out the value of the surface integral
\(\iint_S \bm{F}\cdot\mathrm{d}\bm{S}.\)
Consider the cube of side-length \(\ell\)
in the first quadrant with diagonally opposite corners
located at \((0,0,0)\) and \((\ell,\ell,\ell).\)
The surface \(S\) consists of the three sides of this cube
that lie in the coordinate planes,
and \(\bm{F}(x,y,z) = \langle x,y,z \rangle.\)
The surface \(S\) is the cylinder with height \(h\)
and base of radius \(R\) in the \(xy\)-plane,
and \(\bm{F}(x,y,z) = \bigl\langle x\ln(x^2+y^2), y\ln(x^2+y^2), 0 \bigr\rangle.\)
The surface \(S\) is the sphere of radius \(R\) centered at the origin,
and \[\bm{F}(x,y,z) = \biggl\langle x\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)}, y\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)}, z\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)} \biggr\rangle.\]
The surface \(S\) is the cube of side-length \(\ell\)
in the first quadrant with diagonally opposite corners
located at \((0,0,0)\) and \((\ell,\ell,\ell),\)
and \(\bm{F}(x,y,z) = \langle f(x), 0, 0 \rangle\)
where \(f\) is an abritrary scalar function of \(x\).