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Calculate the Jacobian determinant of the polar coordinate transformation. X=rcosθ Y=rsinθ

Calculate the Jacobian determinant of the polar coordinate transformation. X=rcosθ Y=rsinθ

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 31E
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Calculate the Jacobian determinant of the polar coordinate transformation. X=rcosθ Y=rsinθ
Name...... Lac So for example in 2D and 3D (assuming square matrices) we have: J = of oh oh 8f₂ მ. Oy Or af + J= of u= 3x - 2y v=x+y A = 1 55 55 of Sometimes the determinant of this matrix is also called the Jacobian too. of dz fa fa fa Jy Da One of the meaning's of the Jacobian determinant is as a scaling factor. This will be used a lot in multiple integrals in alternative coordinate systems. Jacobian as a Scaling Factor Let's start by assuming we would like to make a substitution (perhaps in an integral later on): - (9)-(3)() How does this substitution affect the domain in the xy-plane? Consider the unit square with area equal to 1. a fa (x,y) (0,0) (1,0) (0, 1) (1,1) Transforming vertices (u, v) Matrix form (0,0) (3,1) (-2,1) (1,2) .….... ID.…........ Application of Derivative So for example in 2D and 3D (assuming square matrices) we have: oh oh oh oh oh oh By - ( J= of dz ofa Əz A = 1 of By 01₂ Jy Sometimes the determinant of this matrix is also called the Jacobian too. J= One of the meaning's of the Jacobian determinant is as a scaling factor. This will be used a lot in multiple integrals in alternative coordinate systems. Plotting the vertices in the uv-plane shows us how the square domain gets transformed. are: The new area A' is given by: afa Oz w₁ (3,1) = A' = A'= ? 3-2 -|- -|-5 =5 dudv = 5dxdy Wy == w₂ (-2,1) A'=5A In other words the original domain was scaled by a factor of 5 which is equal to the determinant of the vectors in the uv-plane. Advanced Calculu PEARSON Problem 1 Calculate the Jacobian determinant of the polar coordinate transformation. x = r cos 0, y = rsin
Transcribed Image Text:Name...... Lac So for example in 2D and 3D (assuming square matrices) we have: J = of oh oh 8f₂ მ. Oy Or af + J= of u= 3x - 2y v=x+y A = 1 55 55 of Sometimes the determinant of this matrix is also called the Jacobian too. of dz fa fa fa Jy Da One of the meaning's of the Jacobian determinant is as a scaling factor. This will be used a lot in multiple integrals in alternative coordinate systems. Jacobian as a Scaling Factor Let's start by assuming we would like to make a substitution (perhaps in an integral later on): - (9)-(3)() How does this substitution affect the domain in the xy-plane? Consider the unit square with area equal to 1. a fa (x,y) (0,0) (1,0) (0, 1) (1,1) Transforming vertices (u, v) Matrix form (0,0) (3,1) (-2,1) (1,2) .….... ID.…........ Application of Derivative So for example in 2D and 3D (assuming square matrices) we have: oh oh oh oh oh oh By - ( J= of dz ofa Əz A = 1 of By 01₂ Jy Sometimes the determinant of this matrix is also called the Jacobian too. J= One of the meaning's of the Jacobian determinant is as a scaling factor. This will be used a lot in multiple integrals in alternative coordinate systems. Plotting the vertices in the uv-plane shows us how the square domain gets transformed. are: The new area A' is given by: afa Oz w₁ (3,1) = A' = A'= ? 3-2 -|- -|-5 =5 dudv = 5dxdy Wy == w₂ (-2,1) A'=5A In other words the original domain was scaled by a factor of 5 which is equal to the determinant of the vectors in the uv-plane. Advanced Calculu PEARSON Problem 1 Calculate the Jacobian determinant of the polar coordinate transformation. x = r cos 0, y = rsin
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