# Problem Set 19

1. Easy
1. Determine the values of $$\alpha$$ and $$\beta$$ such that $\mathbf F(x,y,z) = (y+z, \alpha x+ z, x+\beta y)$ is a conservative vector field. Write down the potential function $$f: \mathbb R^3 \to \mathbb R$$ so that $$\mathbf F = \nabla f$$.
2. Let $$C$$ be the curve given parametrically by $\gamma(t) = (t\cos(t), t\sin(t), t^2), \qquad 0\leq t \leq \frac\pi 2$ Compute the line integral $$\int_C \mathbf F \cdot d\mathbf x$$ where $$\mathbf F$$ is the your solution from part (1).

2. Easy
Compute the line integral of curve $$\left\{x^2+y^2=1, y\geq 0\right\}$$ (oriented counter clockwise) through the vector field $$\mathbf F(x,y) = (2xy^2+1,2x^2y+2y)$$.

3. Easy
1. Show that the vector field $$\mathbf F(x,y,z) = (y\cos(xy), x\cos(xy)+e^z, ye^z)$$ is conservative and find its scalar potential.
2. Consider the curve $$C$$ given by the intersection of the sets $C = \left\{x^2+y^2+z^2=4\right\} \cap \left\{x=0\right\} \cap \left\{z\geq 0\right\}$ oriented clockwise. Determine $$\int_C \mathbf F \cdot d\mathbf x$$ if $$\mathbf F$$ is the vector field given in part (1).

4. Easy
Evaluate the following line integrals both directly and by using Green's Theorem.
1. $$\displaystyle \oint_C (x+2y) dx + (x-2y) dy$$ where $$C$$ is given by the union of the images of the following two functions on $$[0,1]$$: $$f(x) = x^2$$ and $$g(x) = x$$, positively oriented with respect to the area the curves bound.
2. $$\displaystyle \oint_C (3x-5y) dx + (x+6y) dy$$ where $$C$$ is the ellipse $$x^2/4 + y^2=1$$ oriented counter-clockwise.

5. Medium
1. Let $$D \subseteq \mathbb R^2$$ be a regular region and $$\partial D = C$$ be a piece-wise smooth simple closed curve, oriented positively. If $$A(D)$$ is the area of $$D$$, show that $A(D) = \oint_C x dy = \oint_C -y dx = \oint_C \frac12( -ydx+xdy) .$
2. Consider the disk of radius $$r$$, $$D_r=\left\{x^2+y^2\leq r\right\} \subseteq \mathbb R^2$$. Use any of the formulae from part (1) to compute the area of this disk. [Of course, you already know what the result should be!]
3. In this question we will show that artificially adding boundaries does not affect the line integral. Let $$L_r$$ be any diameter of $$D$$. Show that if we break $$D$$ into two regions $$D = D_1 \cup D_2$$ and give the boundary of $$D_1$$ and $$D_2$$ positive orientations, then for any $$C^1$$ vector field $$\mathbf F(x,y)$$ we have $\oint_{\partial D} \mathbf F \cdot d\mathbf x = \oint_{\partial D_1} \mathbf F \cdot d\mathbf x + \oint_{\partial D_2} \mathbf F\cdot d\mathbf x.$
4. Use part (1) to compute the area of the lemniscate $$x^4=x^2-y^2$$. [Be careful, as the lemniscate's boundary is not a simple closed curve.]

6. Medium
Let $$C$$ be a smooth curve in$$\mathbb R^3$$ parameterized by a vector function $$\mathbf r: [a,b] \to \mathbb R^3$$. Define the identity vector field $$\mathbf F(x,y,z) = (x, y, z)$$
1. Show that $\int_C \mathbf{F} \cdot \mathrm d\mathbf{r} = \frac{1}{2} \left( \vert\vert \mathbf{r}(b)\vert \vert^2 - \vert \vert \mathbf{r}(a)\vert\vert^2 \right) .$
2. What can you conclude about $$\int_C \mathbf F \cdot \mathrm d\mathbf r$$ when $$C$$ is a closed curve?

7. Bonus
For this problem, we will always be working in $$\mathbb R^3$$. Let $$\Omega^0(\mathbb R^3)$$ be the set of $$C^1$$ functions $$\mathbb R^3 \to \mathbb R$$ and $$\Omega^1(\mathbb R^3)$$ be the set of $$C^1$$ vector fields $$\mathbb R^3 \to \mathbb R^3$$.
1. Show that $$\Omega^0(\mathbb R^3)$$ and $$\Omega^1(\mathbb R^3)$$ are both $$\mathbb R$$-vector spaces. [This should be a short proof!]
2. Show that $$\text{grad}: \Omega^0 \to \Omega^1$$ and $$\text{curl}: \Omega^1 \to \Omega^1$$ are linear operators.
3. Recall that if $$Z$$ is the set of closed vector fields, then $$Z = \text{ker}(\text{curl})$$ and if $$B$$ is the set of exact vector fields, then $$B = \text{im}(\text{grad})$$. Show that $$B \subseteq Z$$.
4. If $$S \subseteq \mathbb R^3$$, we define the de Rham cohomology of $$S$$ of degree one to be $H^1(S) = Z/B.$ Show that if $$S$$ is the complement of the $$z$$-axis in $$\mathbb R^3$$ then $$H^1(S)$$ is not the trivial vector space. [Hint: It suffices to show that $$Z \neq B$$.]

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