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Lecture 5: Jacobians In 1D problems we are used to a simple change of variables, e.g. from x to u 1D Jacobian maps strips of width dx to strips of width du Example: Substitute

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**2D Jacobian For a continuous 1-to-1 transformation from (x,y) to (u,v)**

Then Where Region (in the xy plane) maps onto region in the uv plane Hereafter call such terms etc 2D Jacobian maps areas dxdy to areas dudv

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**Why the 2D Jacobian works**

Transformation T yield distorted grid of lines of constant u and constant v For small du and dv, rectangles map onto parallelograms This is a Jacobian, i.e. the determinant of the Jacobian Matrix

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**Relation between Jacobians**

The Jacobian matrix is the inverse matrix of i.e., Because (and similarly for dy) This makes sense because Jacobians measure the relative areas of dxdy and dudv, i.e So

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Simple 2D Example Area of circle A= r

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Harder 2D Example where R is this region of the xy plane, which maps to R’ here 1 4 8 9

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An Important 2D Example Evaluate First consider Put as a -a a -a

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**3D Jacobian maps volumes (consisting of small cubes of volume**

to small cubes of volume Where

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3D Example Transformation of volume elements between Cartesian and spherical polar coordinate systems (see Lecture 4)

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Chapter 5 Multiple integrals; applications of integration

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