M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesChapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple IntegralsLearning module LM 15.1: Multiple integralsLearning module LM 15.2: Multiple integrals over rectangles:Learning module LM 15.3: Double integrals over general regions:Learning module LM 15.4: Double integrals in polar coordinates:Learning module LM 15.5a: Multiple integrals in physics:Learning module LM 15.5b: Integrals in probability and statistics:Learning module LM 15.10: Change of variables:Change of variable in 1 dimensionMappings in 2 dimensions Jacobians Examples Cylindrical and spherical coordinates |
Mappings in 2 dimensionsChanging variables is a very useful technique for simplifying many types of math problems. You can use a horizontal translation f(x)→f(x+1) to change a parabola y=f(x)=x2−2x+1 with vertex at the (1,0) into another parabola y=f(x+1)=x2 with vertex at the origin. We also use changes of variables to convert hard integrals into easier integrals. The change-of-variables formula for ordinary integrals is ∫baf(x)dx = ∫βαf(g(u))g′(u)du,g:[α,β] → [a,b]. Transformations in higher dimensions, called maps or mappings, play an even more important role in multi-variable calculus. We have already seen one such mapping Φ:R2→R2, namely polar coordinates: Φ:(r,θ)→(x,y),x = rcosθ,y = rsinθ, The reason mappings like these are so useful in double integrals comes from their action on particular sets in the plane.
In this case, if D∗=[a,b]×[c,d], then ∫∫Df(x,y)dxdy = ∫ba(∫dcf(Φ(u,v))|∂(x,y)∂(u,v)|dv)du. When the region of integration D in the xy plane has rotational symmetry, polar coordinates often send a rectangle D∗ in the rθ plane to a more complicated region D.
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