# Problem Set 17

1. Easy
Determine the Jacobian of the following transformations. Whenever possible, write the infinitesimal area/volume element in terms of one another:
1. $$(x,y) = (e^{\xi}, \eta^3)$$,
2. $$(x,y) = (5u-2v, u+v)$$,
3. $$(x,y) = (\sin(u^2v), \cos(v^2u))$$,
4. $$(x,y,z)= (v+w^2, w+u^2, u+v^2)$$,
5. $$(x,y,z) = (u^3-v^2, u^3+v^2, u^3+v^2+w)$$.

2. Easy
Let $$R$$ be the region bounded by the curves $$y=x^2$$, $$4y=x^2$$, $$xy=1$$ and $$xy=2$$. Compute the integral $\iint_R x^2 y^2 dx dy.$

3. Easy
Determine $$\displaystyle \iint_S \frac{(x+y)^4}{(x-y)^5} dA$$ where $$S = \left\{ -1 \leq x+y \leq 1,\ 1 \leq x-y \leq 3\right\}$$.

4. Easy
Compute $$\displaystyle\iint_R (4x+8y) dA$$ where $$R$$ is bounded by the quadrilateral with endpoints $$(-1,3), (1,-3), (3,-1), (-3,1)$$.

5. Easy
Compute $$\displaystyle \iint_R \sin(9x^2+4y^2) dA$$ where $$R$$ is bounded by the circle $$9x^2+4y^2=36$$.

6. Easy
Compute $$\displaystyle \iint_R x^2 dA$$ where $$R$$ is bounded by the region $$a^2x^2+b^2y^2=c^2$$, $$a,b,c>0$$.

7. Easy
1. Describe a cone using spherical coordinates.
2. Describe a cone using cylindrical coordinates.
Which of these is more "natural"? Put another way, which of these is "nicer"?

8. Medium
Let $$R\subseteq \mathbb R^2$$ be the region in the $$1^{st}$$ quadrant bounded by the curves $$y=x^2$$, $$y=x^2/5$$, $$xy=2$$, and $$xy=4$$.
1. Define the variables $$u = x^2/y$$ and $$v = xy$$. Compute $$dx dy$$ in terms of $$du dv$$.
2. Compute the area of $$R$$ by changing variables from $$(x,y)$$ to $$(u,v)$$.

9. Medium
1. Let $$R=[a,b]\times[c,d]$$ be a rectangle and $$f: R\subseteq \mathbb R^2 \to \mathbb R$$ be a continuous function. Assume there exist functions $$f_1,f_2: \mathbb R \to \mathbb R$$ such that for all $$x,y \in R$$, $$f(x,y) = f_1(x)f_2(y)$$. Show that $\iint_R f(x,y) dA = \left[\int_{[a,b]} f_1(x) dx \right]\left[\int_{[c,d]} f_2(y) dy\right].$
2. It is known that the function $$f(x) = e^{-x^2}$$ does not have an elementary anti-derivative; however, this function is Riemann integrable on all of $$\mathbb R$$ (by using improper integrals). Compute $\int_{-\infty}^\infty e^{-x^2} dx.$ [Hint: Consider the function $$e^{-x^2-y^2}$$ on $$\mathbb R^2$$ and use part (b)].

10. Medium
Let $$F(\varphi,\theta) = (x(\varphi,\theta),y(\varphi,\theta),z(\varphi,\theta))$$ be the spherical polar coordinate system on the unit sphere $$S^{2} = \left\{(x,y,z) : x^2+y^2+z^2=1\right\}$$.
1. Sketch the coordinate curves $$\theta = \pi/4$$ and $$\varphi = \pi/2$$
2. Compute the derivative to the coordinate curves from part (1) at the point $$(\varphi,\theta) = (\pi/2,\pi/4)$$. Add these arrows to your plot.
3. Prove that $$\partial_{\varphi}F \times \partial_{\theta}F$$ is parallel to $$\nabla(x^2+y^2+z^2)$$.

11. Medium
1. Describe a sphere using cylindrical coordinates
2. Describe a cylinder using spherical coordinates.

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