91Ó°ÊÓ

Find the curl of the vector field \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=(2 y-z) \mathbf{i}+x y z \mathbf{j}+e^{z} \mathbf{k}\)

Verify the Divergence Theorem by evaluating \(\mathbf{F}(x, y, z)=x y \mathbf{i}+z \mathbf{j}+(x+y) \mathbf{k}\) S: surface bounded by the planes \(y=4\) and \(z=4-x\) and the coordinate planes

Find the mass of the surface lamina \(S\) of density \(\rho\). S: \(2 x+3 y+6 z=12\), first octant, \(\rho(x, y, z)=x^{2}+y^{2}\)

Find the mass of the surface lamina \(S\) of density \(\rho\). \(S: z=\sqrt{a^{2}-x^{2}-y^{2}}, \quad \rho(x, y, z)=k z\)

State the Divergence Theorem.

How do you determine if a point \(\left(x_{0}, y_{0}, z_{0}\right)\) in a vector field is a source, a sink, or incompressible?

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field \(\mathbf{F}(x, y, z)\). Find $$\int_{S} \int(\operatorname{curl} \mathbf{F}) \cdot \mathbf{N} d S$$ where \(S\) is the upper surface of the cylindrical container. \(\mathbf{F}(x, y, z)=-z \mathbf{i}+y \mathbf{k}\)

Find the flux of \(F\) through \(S\), \(\iint_{S} \int \mathbf{F} \cdot \mathbf{N} d \boldsymbol{S}\) where \(\mathrm{N}\) is the upward unit normal vector to \(S\). \(\mathbf{F}(x, y, z)=4 \mathbf{i}-3 \mathbf{j}+5 \mathbf{k}\) \(S: z=x^{2}+y^{2}, \quad x^{2}+y^{2} \leq 4\)

Verify that the vector field is conservative. \(\mathbf{F}(x, y)=\frac{1}{x y}(y \mathbf{i}-x \mathbf{j})\)

Use Green's Theorem to verify the line integral formulas. The area of a plane region bounded by the simple closed path \(C\) given in polar coordinates is \(A=\frac{1}{2} \int_{C} r^{2} d \theta\).