Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Related questions
Question
Calculate the Jacobian determinant of
the polar coordinate transformation.
X=rcosθ
Y=rsinθ
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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