2. First, read this definition. The Jacobian of a transformation T given by x = g(u, v) and y = h(u, v) is (x,y) (u, v) =xuyvxvYu - Assuming the transformation T is C¹ (meaning g and h have continuous partial derivatives) and T is a one to one transformation from a region R in the xy-plane to a region S in the uv-plane, then √ f(x, y)dA = f(z(u, v), y(u, v)) | [] (x,y) |(u, v) dudv Note the absolute value around the Jacobian) This is how we can do a substitution in a double integral! Now answer the following questions. (a) If R is the square with vertices (0,2), (1, 1), (2, 2), (1,3) and u = xy, v = x + y, sketch the new region in the uv-plane. Be sure to label your graph and provide coordinates of all vertices. 1 (b) Now find the Jacobian of the transformation (c) Now use the information above to integrate the following double integral. R - Y dA x + y
2. First, read this definition. The Jacobian of a transformation T given by x = g(u, v) and y = h(u, v) is (x,y) (u, v) =xuyvxvYu - Assuming the transformation T is C¹ (meaning g and h have continuous partial derivatives) and T is a one to one transformation from a region R in the xy-plane to a region S in the uv-plane, then √ f(x, y)dA = f(z(u, v), y(u, v)) | [] (x,y) |(u, v) dudv Note the absolute value around the Jacobian) This is how we can do a substitution in a double integral! Now answer the following questions. (a) If R is the square with vertices (0,2), (1, 1), (2, 2), (1,3) and u = xy, v = x + y, sketch the new region in the uv-plane. Be sure to label your graph and provide coordinates of all vertices. 1 (b) Now find the Jacobian of the transformation (c) Now use the information above to integrate the following double integral. R - Y dA x + y
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![2. First, read this definition.
The Jacobian of a transformation T given by x = g(u, v) and y = h(u, v) is
(x,y)
(u, v)
=xuyvxvYu
-
Assuming the transformation T is C¹ (meaning g and h have continuous partial derivatives) and T is a
one to one transformation from a region R in the xy-plane to a region S in the uv-plane, then
√ f(x, y)dA = f(z(u, v), y(u, v)) |
[]
(x,y)
|(u, v)
dudv
Note the absolute value around the Jacobian)
This is how we can do a substitution in a double integral!
Now answer the following questions.
(a) If R is the square with vertices (0,2), (1, 1), (2, 2), (1,3) and u = xy, v = x + y, sketch the new
region in the uv-plane. Be sure to label your graph and provide coordinates of all vertices.
1
(b) Now find the Jacobian of the transformation
(c) Now use the information above to integrate the following double integral.
R
-
Y
dA
x + y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe55cfba9-7bce-45cb-a868-b9c474dd1b77%2Fdebdaf5f-f2e9-4104-b917-bf2bfe3493cf%2Fpc0mh2h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. First, read this definition.
The Jacobian of a transformation T given by x = g(u, v) and y = h(u, v) is
(x,y)
(u, v)
=xuyvxvYu
-
Assuming the transformation T is C¹ (meaning g and h have continuous partial derivatives) and T is a
one to one transformation from a region R in the xy-plane to a region S in the uv-plane, then
√ f(x, y)dA = f(z(u, v), y(u, v)) |
[]
(x,y)
|(u, v)
dudv
Note the absolute value around the Jacobian)
This is how we can do a substitution in a double integral!
Now answer the following questions.
(a) If R is the square with vertices (0,2), (1, 1), (2, 2), (1,3) and u = xy, v = x + y, sketch the new
region in the uv-plane. Be sure to label your graph and provide coordinates of all vertices.
1
(b) Now find the Jacobian of the transformation
(c) Now use the information above to integrate the following double integral.
R
-
Y
dA
x + y
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