Line integral

Exercise3.3

1.

Given the force field ${\mathbf F} = 3xy \boldsymbol{i} - 5{\bf z} + 10 x\boldsymbol{k}$. Find the work done by ${\mathbf F}$ which is $W = \int_{C}{\mathbf F}\cdot d\boldsymbol{r}$, moving along the curve

$$C : x = t^2 + 1,\ y = 2t^2,\ z = t^3$$

from $t = 1$ to $t = 2$.

2.

Given the scalar field $\phi = 2xyz^2$，the field ${\mathbf F} = xy\boldsymbol{i} - z\boldsymbol{j} + x^2 \boldsymbol{k}$. Let the curve $C$ be parametrized by $x = t^2, y = 2t, z = t^3\ (0 \leq t \leq 1)$．Find the following line integrals.

(1)

$\int_{C}\phi d\boldsymbol{r}$(2) $\int_{C}{\mathbf F} \times d\boldsymbol{r}$

3.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z \boldsymbol{k}$．Then for any closed curve $C$, prove that $\int_{C}\boldsymbol{r}\cdot d\boldsymbol{r} = 0$．

4.

Suppose that the force field ${\mathbf F}$ has the potential $U$．Within the force field, the point mass with the mass $m$ moves from the point A to the point B, show that following equation holds．

$$\frac{1}{2}mv_{A}^2 + U(A) = \frac{1}{2}mv_{B}^2 + U(B)$$

Here， $v_{A},v_{B}$ are the magnitude of velocity vectors of A and B.

5.

$\Phi = \tan^{-1}{\frac{y}{x}}$ is defined in the domain $D$ excluding the $z$ axis from the whole space． Let $C$ be a circle with a radius of $a$ centered at the origin O on the $xy$ plane. Prove the following equation.

$$\int_{C}(\nabla \phi)\cdot d\boldsymbol{r} = 2\pi$$

#F#>ind the amount of work done by the mass point to go around $${\frac{x^2}{16} + \frac{y^2}{9} = 1}$$. Note that vector field is given by

$$\boldsymbol{F} = (3x-4y+2z)\:\boldsymbol{i} + (4x+2y-3z^2)\:\boldsymbol{j} + (2xz-4y^2 + z^3)\:\boldsymbol{k}$$

7.

Evaluate the line integral $${\int_{C}(x^3 + y^4)ds}$$, where $C$ is a line connecting the points $(0,0,0)$ and $(1,1,1)$.

8.

Find the line integral $${\int_{C}(x^2 y\boldsymbol{i} + (x+y)\:\boldsymbol{j}) \cdot{\bf t}ds}$$．Here， $C$ is the curve $${y = x^2}$$ from $(0,0)$ to $(2,4)$.