Calculus 3 Change of Variables: Jacobians

Substitute that into u= y/x: u= y/(v/(u2+1)). Multiply both sides by v. uv= y(u2+1). Solve for y. y= uv/(u2+1).In summary, we must evaluate the integral \int\inte^xy dA, where R is the region enclosed by the curves y/x=1/2, y/x=2, xy=1, and xy=2. To do so, we can use the change of variables functions u=y/x and v=xy, which result in the equations x=v/(u2+1) and y=uv/(u2+1).
  • #1
Wargy
3
0

Homework Statement


Evaluate [tex]\int[/tex][tex]\int[/tex]e^xy dA, where R is the region enclosed by the curves: y/x=1/2 , y/x=2, xy=1, and xy=2.


Homework Equations


None?


The Attempt at a Solution


I have the region graphed and I'm currently working on acquiring the change of variables functions in x and y. I have attempted to solve the system of equations with u=y/x and v=xy to obtain these but I'm having some trouble. If I could be pointed in the right direction I would be greatly appreciative! Thanks.
 
Physics news on Phys.org
  • #2
You have u= y/x and v= xy so y= xu. Substitute that into v= xy: v= x2u. Now solve for x.
 

FAQ: Calculus 3 Change of Variables: Jacobians

What is a Jacobian in Calculus 3?

The Jacobian is a matrix of partial derivatives used to transform variables in a multivariable function. It is used to convert integrals in one coordinate system to integrals in another coordinate system.

Why is the Jacobian important in Change of Variables?

The Jacobian is important in Change of Variables because it helps to simplify and solve integrals in multiple dimensions. By transforming the variables, it allows for a more efficient and accurate calculation of integrals.

How do you calculate the Jacobian in Calculus 3?

The Jacobian can be calculated by taking the partial derivatives of each variable with respect to each other. For example, if we have a function f(x,y,z) and want to transform it to a new variable system u,v,w, the Jacobian would be the matrix [∂x/∂u ∂x/∂v ∂x/∂w; ∂y/∂u ∂y/∂v ∂y/∂w; ∂z/∂u ∂z/∂v ∂z/∂w].

Can the Jacobian be used for any type of coordinate system?

Yes, the Jacobian can be used for any type of coordinate system, including polar, cylindrical, and spherical coordinates. It is a versatile tool for transforming variables in multivariable functions.

What is the relationship between the Jacobian and the Change of Variables theorem?

The Change of Variables theorem states that when performing a change of variables for an integral, the Jacobian must be factored into the integral to ensure the correct transformation. In other words, the Jacobian is a necessary component in applying the Change of Variables theorem.

Similar threads

Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K
Change of variable for Jacobian: is there a method?
Replies
5
Views
1K
Calculate the joint CDF of two random variables
Replies
2
Views
2K
Calculus 3 change of variables
Replies
3
Views
1K
Determine limits of integration in double integral change of variables
Replies
2
Views
960
Find the dimensions that will minimize the surface area of a Rectangle
Replies
1
Views
871
Changing Variables and the Limits of Integration using the Jacobian
Replies
21
Views
2K
Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##
Replies
5
Views
946
Double Integral via Appropriate Change of Variables
Replies
3
Views
1K
Problem solving this volume using Jacobi's Determinant
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top