91Ó°ÊÓ

Evaluate \(\iint_{s} g(x, y, z) d S\) \(g(x, y, z)=x^{2}\) \(S\) is the upper half of the sphere \(x^{2}+y^{2}+z^{2}=a^{2} .\)

Use the divergence theorem (18.26) to find \(\iint_{s} F \cdot \mathbf{n} d S\) \(\mathbf{F}=y \sin x \mathbf{i}+y^{2} z \mathbf{j}+(x+3 z) \mathbf{k}\) \(S\) is the surface of the region bounded by the planes \(x=\pm 1, y=\pm 1, z=\pm 1\)

Show that \(\int_{c} \mathbf{F} \cdot d \mathbf{r}\) is independent of path by finding a potential function \(f\) for \(F\). $$ \mathbf{F}(x, y)=\left(3 x^{2} y+2\right) \mathbf{i}+\left(x^{3}+4 y^{3}\right) $$

Use Green's theorem to evaluate the line integral. \(\oint_{c}\left(x^{2}+y\right) d x+\left(x y^{2}\right) d y\) \(C\) is the closed curve determined by \(y^{2}=x\) and \(y=-x\) with \(0 \leq x \leq 1\)

Sketch some typical vectors in the vector field F. $$ \mathbf{F}(x, y)=2 x \mathbf{i}+y \mathbf{j} $$

Sketch a sufficient number of vectors to illustrate the pattern of the vectors in the field \(F\). $$ \mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j} $$

Exer. \(1-2:\) Evaluate the line integrals \(\int_{C} f(x, y) d s\) \(\int_{\boldsymbol{C}} f(x, y) d x,\) and \(\int_{c} f(x, y) d y\) if \(C\) has the given parametrization. \(f(x, y)=x^{3}+y, \quad x=3 t, \quad y=t^{3} ; \quad 0 \leq t \leq 1\)

Use the divergence theorem (18.26) to find \(\iint_{s} F \cdot \mathbf{n} d S\) \(\mathbf{F}=y^{3} e^{z} \mathbf{i}-x y \mathbf{j}+x \arctan y \mathbf{k}\) \(S\) is the surface of the region bounded by the coordinate planes and the plane \(x+y+z=1\).

Exer. 1-4: Verify Stokes' theorem (18.28) for \(F\) and \(S\). \(\mathbf{F}=2 y \mathbf{i}-z \mathbf{j}+3 \mathbf{k}\) \(S\) is the portion of the paraboloid \(z=4-x^{2}-y^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=1\)

Sketch some typical vectors in the vector field F. $$ \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+\mathbf{k} $$