GM Week 7

GM Week 7

1. Cartesian Space

   • R^2: dA = dx dy
   • R^3: dA = dxdydz

2. Polar Coordinates

   • x = r cos θ
   • y = r sin θ
   • r^2 = x^2 + y^2
   • tan θ = y/x
   • • dA = r dr dθ

3. Cylindrical Coordinates

   • dV = rdrdθd

4. Spherical Coordinates

   • φ is the angle from the vertical (angle from the north pole or co-latitude)
   • x = r sin φ cos θ
   • r = (x^2 + y^2 + z^2 )^1/2
   • y = r sin φ sin
   • z = r cos φ
   • tan φ = ((x^2 + y^2)^1/2)/z
   • tanθ=y/x
   • 0 ≤ φ ≤ π
   • 0 ≤ θ ≤ 2π
   • dV = r^2 sin φ dr dθ dφ

5. Discussion

   • Sometimes we use special coordinates because of geometry, that is the region of integration suggest special coordinates, but equally often we use special coordinates because the integrand simplifies in these coordinates. Another way to say this is that in many cases the function we wish to integrate arises because a problem depends only on distance from the origin (polar coordinates in 2D and spherical coordinates in 3D) or because a problem depends only on distance from an axis (cylindrical coordinates).
   • In problems with symmetry, the integral separates in the corresponding coordinate, or coordinates, and the integrals simplify.
   • Recall Type I and Type II regions for double integral for example. In general, in iterated integration the limits of any inner integrals may depend on coordinates of any of the outer integrals.

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