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Exercise 4. A transformation you should have some familiarity with already is the transformation from Cartesian coordinates (x, y) to polar coordinates (r,0) given by: x = rcos(0) y = rsin(0) Using the theory developed above, obtain the well-known relationship between differential areas for the two coordinate systems: dædy = rdrd0

Exercise 4. A transformation you should have some familiarity with already is the transformation from Cartesian coordinates (x, y) to polar coordinates (r,0) given by: x = rcos(0) y = rsin(0) Using the theory developed above, obtain the well-known relationship between differential areas for the two coordinate systems: dædy = rdrd0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 98E
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2.2 The General Case Definition 2. For a differentiable transformation T: Rm → R", the Jacobian matrix is the m x m matrix J so that Jij = (10) Note that for a linear transformation the Jacobian is simply the matrix of the transformation. (If you have a little trouble seeing the general case, look at the particular case of Equation (4).) Using the idea that differentiable transformations are locally linear, one obtains the differential identity du du2 - - - dum = |J(x)| dx1dx2- - dxm (11) One must be a little careful in getting the direction of the transformation right ... However, assuming that the inverse of the transformation exists at any point u, one has: |J(u)| = |J(x)|-1 (12)
Transcribed Image Text:2.2 The General Case Definition 2. For a differentiable transformation T: Rm → R", the Jacobian matrix is the m x m matrix J so that Jij = (10) Note that for a linear transformation the Jacobian is simply the matrix of the transformation. (If you have a little trouble seeing the general case, look at the particular case of Equation (4).) Using the idea that differentiable transformations are locally linear, one obtains the differential identity du du2 - - - dum = |J(x)| dx1dx2- - dxm (11) One must be a little careful in getting the direction of the transformation right ... However, assuming that the inverse of the transformation exists at any point u, one has: |J(u)| = |J(x)|-1 (12)
Exercise 4. A transformation you should have some familiarity with already is the transformation from Cartesian coordinates (x, y) to polar coordinates (r,0) given by: x = rcos(0) y = rsin(0) Using the theory developed above, obtain the well-known relationship between differential areas for the tuwo coordinate systems: dædy = rdrd0
Transcribed Image Text:Exercise 4. A transformation you should have some familiarity with already is the transformation from Cartesian coordinates (x, y) to polar coordinates (r,0) given by: x = rcos(0) y = rsin(0) Using the theory developed above, obtain the well-known relationship between differential areas for the tuwo coordinate systems: dædy = rdrd0
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