Change of Variable

    With a function of one variable we often use a change of variable (a substitution) to simplify an integral. By reversing the roles of and an integral can be rewritten,



where and This factor is crucial for determining whether a change of variable will be successful, or not. In order to have a change of variables for functions of two or more variables, a factor which is defined in terms of partial derivatives will be needed.

Definition (Change of Variable) If and then the Jacobian of and with respect to and denoted by is  



More generally, for



the Jacobian of with respect to is  

Example (Change of Variable) Compute the Jacobian for the conversion from the rectangular plane to the polar plane.

    Solution. The conversion formulas are and So the Jacobian is,
    

Example (Change of Variable) Compute the Jacobian for the conversion from rectangular coordinates to cylindrical coordinates.

    Solution.  The conversion formulas are and So the Jacobian is,
    

Example (Change of Variable) Compute the Jacobian for the conversion from rectangular coordinates to spherical coordinates.

    Solution. The conversion formulas are and So the Jacobian is,
    

Proposition (Change of Variables in Double Integrals) Let be a continuous function on a region in the and let be a one-to-one transformation that maps the region in the -plane onto under the change of variables where and are continuously differentiable functions on If   then

Example (Change of Variable) Use a change of variables to compute the area of an ellipse.

    Solution. Assume the ellipse is given in standard form by Let and then the ellipse in the -plane corresponds to the unit circle in the -plane. Since and the Jacobian is and so the area of an ellipse is given by  
    

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the trapezoidal region with vertices and
    
    Solution. Since it is not easy to integrate we make a change of variables suggested by the form of namely: and   Then, since


        
the Jacobian is,



To find the region in the -plane corresponding to we note that the sides of lie on the lines and using the rules for the transformation, and the images of the lines in the are



Thus the region is the trapezoidal region with vertices and that is



The transformation from to is,




Therefore,

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the region bounded by the lines and
    
    Solution. Let and Then solving for and produces

and

We compute the Jacobian of and



The bounds under the transformation are



Let's consider the region in the -plane as vertically simple as follows,

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the region bounded by the square with vertices and
    
    Solution. The region is bounded by the lines and Let and then solving for and produces

and
    
    We compute the Jacobian of and



The bounds under the transformation are



Let's consider the region in the -plane as horizontally simple as follows,

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the region bounded by the parallelogram with vertices and
    
    Solution. The boundary lines of the parallelogram are and Let and with boundary lines and Solving for and produces

and
    
Since the Jacobian is
    


we have

    In the next example we will make use of



because it is easier not to solve for and in terms of and

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the region bounded by the hyperbolas and
    
    Solution. Let and Then



Since, we have



Thus,



and so



Now then,

Example (Change of Variable) Use a change of variables to evaluate the integral



where is the triangular region bounded by the vertices
    
    Solution. Let and so that and Then the Jacobian is
    


The given the region is bounded by the lines and which transform into and Therefore,

Proposition (Change of Variables in Triple Integrals) Let be a continuous function on a region in the and let be a one-to-one transformation that maps the region in the -space onto under the change of variables and where functions and are continuously differentiable functions on If   then





Example (Change of Variable) Compute the volume of an ellipsoid.

    Solution. Assume the ellipsoid is given in standard form by



Let and then the ellipsoid corresponds to the unit sphere Since the Jacobian is
    


and so the volume of an ellipsoid is given by  
    

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