Math 344: Calculus III |
1. Evaluate the integral \(\displaystyle \int_C x\, ds\) where \(C\) is
the arc of the parabola \(y = x^2\) from \((1,1)\) to \((0,0)\).
2. Evaluate the integral \(\displaystyle \int_C y^3 \, dx + x^2 \, dy\)
where \(C\) is the arc of the parabola \(x = 1-y^2\) from \((0,-1)\) to
\((0,1)\).
3. Compute \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\) where
\(\mathbf{F}(x,y,z) = \langle e^z,xz,(x+y)\rangle\) and \(C\) is given by
\(\mathbf{r}(t) = \langle t^2,t^3,-t\rangle\).
4. Show that \(\mathbf{F}\) is a conservative vector field, then find a
potential function, \(f\).
\[\mathbf{F}(x,y) = (1+xy)e^{xy}\mathbf{i} + (e^y + x^2e^{xy})\mathbf{j}\]
5. Show that \(\mathbf{F}\) is conservative and use that fact to evaluate
\(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\).
\[\begin{cases}
\mathbf{F}(x,y,z) = \langle e^y,xe^y+e^z,ye^z\rangle, \\
C \mathrm{\ is\ the\ line\ segment\ from\ } (0,2,0)\ \mathrm{to\ } (4,0,3).
\end{cases}\]
6. Verify Green's Theorem is true for the line integral
\[\int_C xy^2\, dx - x^2y\, dy \]
where \(C\) consists of the parabola \(y = x^2\) from \((-1,1)\) to \((1,1)\)
and the line segment from \((1,1)\) to \((-1,1)\).
7. Use Green's Theorem to evaluate the path integral, where \(C\) is the
positively oriented triangle with vertices \((0,0), (1,0),\) and \((1,3)\).
\[\displaystyle \int_C \sqrt{1 + x^3}\,dx + 2xy\, dy \]
8. Find \(\mathrm{curl\ }\mathbf{F}\) and \(\mathrm{div\ }\mathbf{F}\) for
\[\mathbf{F}(x,y,z) = e^{-x}\sin y\, \mathbf{i} + e^{-y}\sin z\, \mathbf{j} +
e^{-z}\sin x\, \mathbf{k}. \]
9. Suppose \(f\) is a harmonic function: \(\Delta f = 0\). Show that
\[ \displaystyle \int_C \dfrac{\partial f}{\partial y}\, dx - \dfrac{\partial
f}{\partial x}\, dy = 0 \]
where \(C\) is any simple closed curve in the domain of \(f\).
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