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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