91Ó°ÊÓ

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. \(\int_{C}(y+2 z) d x+(x-3 z) d y+(2 x-3 y) d z\) (a) \(C\) : line segment from \((0,0,0)\) to \((1,1,1)\) (b) \(C\) : line segments from \((0,0,0)\) to \((0,0,1)\) to \((1,1,1)\) (c) \(C:\) line segments from \((0,0,0)\) to \((1,0,0)\) to \((1,1,0)\) to \((1,1,1)\)

Tangent Plane, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. $$ \mathbf{r}(u, v)=(u+v) \mathbf{i}+(u-v) \mathbf{j}+v \mathbf{k}, \quad(1,-1,1) $$

Define a surface integral of the scalar function \(f\) over a surface \(z=g(x, y)\). Explain how to evaluate the surface integral.

Evaluate $$\int_{C} \mathbf{F} \cdot d \mathbf{r}$$ where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}\) C: \(\mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}, \quad 0 \leq t \leq \pi\)

Determine if the vector field is conservative. \(\mathbf{F}(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}(x \mathbf{i}+y \mathbf{j})\)

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 \(A=\frac{1}{2} \int_{C} r^{2} d \theta\).

Tangent Plane, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. $$ \mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+\sqrt{u v} \mathbf{k}, \quad(1,1,1) $$

Describe an orientable surface.

Define a flux integral and explain how it is evaluated.

Use a computer algebra system to evaluate the integral $$\int_{C} \mathbf{F} \cdot d \mathbf{r}$$ where \(C\) is represented by \(\mathbf{r}(t)\). \(\mathbf{F}(x, y, z)=x^{2} z \mathbf{i}+6 y \mathbf{j}+y z^{2} \mathbf{k}\) \(\quad C: \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\ln t \mathbf{k}, \quad 1 \leq t \leq 3\)