# Problem Set 18

1. Easy
Plot the following vector fields:
1. $$F(x,y) = (\sqrt x, y)$$,
2. $$F(x,y) = (y,x)$$,
3. $$F(x,y) = \left( \frac1x, y\right)$$,
4. $$F(x,y) = \left(\frac yx, y\right)$$,
5. $$F(x,y) = (x+y,x-y)$$.

2. Easy
For each of the following functions, compute the gradient, Laplacian, divergence, and curl, if it makes sense to do so.
1. $$F(x,y) = \sin(xe^y)$$,
2. $$F(x,y) = x^2+y^2-2z^2$$,
3. $$F(x,y,z) = (\sin^2(x), \cos^2(z), xyz)$$,
4. $$F(x,y,z) = \frac{ x}{x^2+y^2}$$,
5. $$F(x,y) = \log(x^2+y^2)$$,
6. $$F(x,y,z) = (2xy^2z^2, 2yx^2z^2,2zx^2y^2)$$,
7. $$F(x,y,z,w) = (e^{zw}, e^{xz}, e^{xw}, e^{yz})$$.

3. Easy
Show that the following identities hold:
1. $$\nabla(fg) = f\nabla g + g\nabla f$$
2. $$\text{curl}(f\mathbf G) = f \text{curl} \mathbf G + (\nabla f)\times \mathbf G$$
3. $$\text{div}(f\mathbf G) = f \text{div} \mathbf G + (\nabla f) \cdot \mathbf G$$
4. $$\text{div}(\mathbf F\times \mathbf G) = \mathbf G \cdot (\text{curl} \mathbf F) - \mathbf F \cdot (\text{curl}\mathbf G)$$

4. Easy
Show that the following identities hold:
1. $$\text{curl}(\nabla f) = 0$$
2. $$\text{div}(\text{curl} \mathbf F) = 0$$

5. Easy
Find the arclength of the following curves
1. The straight line between $$(1,2,3)$$ and $$(3,1,2)$$,
2. The curve given by $$y^2=x^3$$ between $$(1,1)$$ and $$(4,8)$$,
3. The curve given by $$(x,y) = (t-\sin(t),1-\cos(t))$$ for $$0 \leq t \leq 2\pi$$,
4. The curve given by $$(x,y) = (\cos^3(t),\sin^3(t))$$ for $$0\leq t\leq \frac\pi2$$,
5. The graph of a $$C^1$$ function $$y=f(x)$$ for $$a \leq x \leq b$$.

6. Easy
Let $$\mathbf F(x,y) = (-y^2,xy)$$ and $$C = \displaystyle\left\{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1: y \geq 0\right\}$$. Determine $$\displaystyle \int_C \mathbf F \cdot d\mathbf x$$ if $$C$$ is oriented counter clockwise when viewed from $$(0,0,1)$$.

7. Easy
Determine $$\int_C \mathbf F \cdot dx$$ where $$\mathbf F(x,y) = (x^2,-y)$$ and $$C$$ is the graph of $$y=e^x$$ from $$(2,e^2)$$ to $$(0,1)$$.

8. Easy
Determine $$\int_C \mathbf F \cdot dx$$ where $$\mathbf F(x,y,z) = (z,-y,x)$$ and $$C$$ is the line segment between the points $$(5,0,2)$$ and $$(5,3,4)$$.

9. Easy
Let $$\mathbf F(x,y,z) = (x,y,z^2)$$ and $$C$$ be the curve given by the intersection of the cylinder $$x^2+y^2=1$$ and $$z=x$$, with any orientation. Determine $$\oint_C \mathbf F \cdot dx$$.

10. Easy
By explicitly parameterizing, show that if $$C$$ is the constant curve (that is, the curve which consists of a single point), then for any vector field $$\mathbf F$$ we have $\int_C \mathbf F \cdot dx = 0.$

11. Easy
Consider a vertical line segment $$S=\left\{(x,y): a \leq y \leq b, x=c\right\}$$ in $$\mathbb R^2$$, for constants $$a,b,c$$. Show that for any vector field $$\mathbf F(x,y) = (F_1(x,y), F_2(x,y)$$ that the line integral does not depend on $$F_1$$. Similarly conclude that for a horizontal line segment, the line integral does not depend on $$F_2$$.

12. Medium
Let $$(x,y)$$ be the standard Cartesian coordinates in $$\mathbb R^2$$. Changing to polar coordinates we set $x = r\cos(\theta), \quad y = r\sin(\theta).$
1. Let $$e_x = (1,0)$$ and $$e_y = (0,1)$$ be the standard Cartesian unit vectors of $$\mathbb R^2$$. Let us define $$e_r=\cos(\theta) e_x + \sin(\theta) e_y$$ and $$e_\theta=-\sin(\theta) e_x + \cos(\theta) e_y$$, to be the standard polar unit ''vectors" (at the point $$(r, \theta$$)).
2. Using the multivariable chain rule, determine the $$\nabla$$ operator in polar coordinates.
3. Compute the Laplacian $$\nabla^2$$ in polar coordinates.

13. Medium
Let $$\mathbf F(x,y,z) = (yz, xz, xy)$$ and define $C_{r,h} = \left\{ (x,y,z): x^2+y^2=r^2, z = h\right\}.$ Show that for and $$r >0$$ and $$h \in \mathbb R$$, $\int_{C_{r,h}} \mathbf F \cdot dx = 0.$

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