91Ó°ÊÓ

Find the flux of field \(\mathbf{F}=-x \mathbf{i}+y \mathbf{j}\) across \(x^{2}+y^{2}=16\) oriented in the counterclockwise direction.

Let \(\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}, \quad\) and let \(C\) be a triangle bounded by \(y=0, x=3,\) and \(y=x\) oriented in the counterclockwise direction. Find the outward flux of \(\mathbf{F}\) through \(C\).

[T] Let \(C\) be unit circle \(x^{2}+y^{2}=1\) traversed once counterclockwise. \(\quad\) Evaluate \(\int_{C}\left[-y^{3}+\sin (x y)+x y \cos (x y)\right] d x+\left[x^{3}+x^{2} \cos (x y)\right] d y\) by using a computer algebra system.

[T] Find the outward flux of vector field \(\mathbf{F}=x y^{2} \mathbf{i}+x^{2} y \mathbf{j}\) across the boundary of annulus \(R=\left\\{(x, y): 1 \leq x^{2}+y^{2} \leq 4\right\\}=\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}\) using a computer algebra system.

Consider region \(R\) bounded by parabolas \(y=x^{2}\) and \(x=y^{2}\). Let \(C\) be the boundary of \(R\) oriented counterclockwise. Use Green's theorem to evaluate \(\oint_{C}\left(y+e^{\sqrt{x}}\right) d x+\left(2 x+\cos \left(y^{2}\right)\right) d y\).

Determine whether the statement is true or false. If the coordinate functions of \(\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) have continuous second partial derivatives, then \(\operatorname{curl}(\operatorname{div}(\mathbf{F}))\) equals zero.

Determine whether the statement is true or false. \(\nabla \cdot(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})=1\)

Determine whether the statement is true or false. All vector fields of the form \(\mathbf{F}(x, y, z)=f(x) \mathbf{i}+g(y) \mathbf{j}+h(z) \mathbf{k}\) are conservative.

Determine whether the statement is true or false. If \(\mathbf{F}\) is a constant vector field then \(\operatorname{div} \mathbf{F}=0\)

Determine whether the statement is true or false. If \(\mathbf{F}\) is a constant vector field then \(\operatorname{curl} \mathbf{F}=0\).