91Ó°ÊÓ

If you have a CAS that plots vector fields (the command is fieldplot in Maple and PlotVectorField in Mathematica), use it to plot $$\mathbf{F}(x, y)=\left(y^{2}-2 x y\right) \mathbf{i}+\left(3 x y-6 x^{2}\right) \mathbf{j}$$ Explain the appearance by finding the set of points \((x, y)\) such that \(\mathbf{F}(x, y)=\mathbf{0} .\)

Is there a vector field \(\mathbf{G}\) on \(\mathbb{R}^{3}\) such that curl \(\mathbf{G}=\langle x \sin y, \cos y, z-x y\rangle ?\) Explain.

\(19-26\) Find a parametric representation for the surface. The plane that passes through the point \((1,2,-3)\) and contains the vectors \(\mathbf{i}+\mathbf{j}-\mathbf{k}\) and \(\mathbf{i}-\mathbf{j}+\mathbf{k}\)

\(19-20\) Show that the line integral is independent of path and evaluate the integral. $$\int_{C} \tan y d x+x \sec ^{2} y d y$$ \(C\) is any path from \((1,0)\) to \((2, \pi / 4)\)

\(19-22\) Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is given by the vector function \(\mathbf{r}(t) .\) $$ \begin{array}{l}{\mathbf{F}(x, y)=x y \mathbf{i}+3 y^{2} \mathbf{j}} \\\ {\mathbf{r}(t)=11 t^{4} \mathbf{i}+t^{3} \mathbf{j}, \quad 0 \leqslant t \leqslant 1}\end{array} $$

If \(S\) is a sphere and \(\mathbf{F}\) satisfies the hypotheses of Stokes' Theorem, show that \(\iint_{S}\) curl \(\mathbf{F} \cdot d \mathbf{S}=0\)

\(19-30\) Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot d S\) for the given vector field \(\mathbf{F}\) and the oriented surface \(S .\) In other words, find the flux of \(\mathbf{F}\) across \(S .\) For closed surfaces, use the positive (outward) orientation. $$\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k}, \quad $$ S is the part of the paraboloid \(z=4-x^{2}-y^{2}\) that lies above the square \(0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1,\) and has upward orientation

\(19-20\) Show that the line integral is independent of path and evaluate the integral. $$\int_{C}\left(1-y e^{-x}\right) d x+e^{-x} d y$$ \(C\) is any path from \((0,1)\) to \((1,2)\)

\(19-22\) Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is given by the vector function \(\mathbf{r}(t) .\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+(y-z) \mathbf{j}+z^{2} \mathbf{k}} \\ {\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+t^{2} \mathbf{k}, \quad 0 \leqslant t \leqslant 1}\end{array} $$

\(19-26\) Find a parametric representation for the surface. The lower half of the ellipsoid \(2 x^{2}+4 y^{2}+z^{2}=1\)