(For those who have studied multivariable calculus. ) Let T be an invertible linear transformation from ℝ 2 to ℝ 2 , represented by the matrix M. Let Ω 1 be the unit square in ℝ 2 and Ω 2 its image under T. Consider a continuous function f ( x , y ) from ℝ 2 to ℝ , and define the function g ( u , v ) = f ( T ( u , v ) ) . What is the relationship between the following two double integrals ? ∬ Ω 2 f ( x , y ) d A and ∬ Ω 1 g ( u , v ) d A Your answer will involve the matrix M. Hint: What happens when f ( x , y ) = 1 , for all x, y?
(For those who have studied multivariable calculus. ) Let T be an invertible linear transformation from ℝ 2 to ℝ 2 , represented by the matrix M. Let Ω 1 be the unit square in ℝ 2 and Ω 2 its image under T. Consider a continuous function f ( x , y ) from ℝ 2 to ℝ , and define the function g ( u , v ) = f ( T ( u , v ) ) . What is the relationship between the following two double integrals ? ∬ Ω 2 f ( x , y ) d A and ∬ Ω 1 g ( u , v ) d A Your answer will involve the matrix M. Hint: What happens when f ( x , y ) = 1 , for all x, y?
Solution Summary: The author explains the relationship between integrals, displaystyleundersetOmega_2iintf(x,y)dA, and
(For those who have studied multivariable calculus.) Let T be an invertible linear transformation from
ℝ
2
to
ℝ
2
, represented by the matrix M. Let
Ω
1
be the unit square in
ℝ
2
and
Ω
2
its image under T. Consider a continuous function
f
(
x
,
y
)
from
ℝ
2
to
ℝ
, and define the function
g
(
u
,
v
)
=
f
(
T
(
u
,
v
)
)
. What is the relationship between the following two double integrals?
∬
Ω
2
f
(
x
,
y
)
d
A
and
∬
Ω
1
g
(
u
,
v
)
d
A
Your answer will involve the matrix M. Hint: What happens when
f
(
x
,
y
)
=
1
, for all x, y?
Study of calculus in one variable to multiple variables. The typical operations involved in multivariate calculus are limits and continuity, partial differentiation, and multiple integration. Major applications are in regression analysis, in finance by quantitative analysis, in engineering and social science to study and build high dimensional systems and exhibit deterministic nature.
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY