# MA 227 Standard S08

Transformations of Variables

At the end of the course, each student should be able to…

• S08: TransVar. Compute and apply a transformation of variables.

## S08: Transformations of Variables

• A transformation of variables is a function $$\vect T(u,v)=\<x,y\>$$ that converts vectors $$\<u,v\>$$ in the $$uv$$ plane into vectors $$\<x,y\>$$ in the $$xy$$ plane.
• A transformation is called affine if it preserves parallelograms.
• All affine transformations are of the form $$\vect T(u,v)=\<a_1u+b_1v+c_1,a_2u+b_2v+c_2\>$$.
• To find a transformation from the unit square in the $$uv$$ plane to a parallelogram, the values of $$a,b,c$$ may be calculated by setting each of $$\vect T(0,0),\vect T(1,0), \vect T(1,1), \vect T(0,1)$$ to each of its corners.
• To find a transformation from the unit triangle in the $$uv$$ plane to a parallelogram, the values of $$a,b,c$$ may be calculated by setting each of $$\vect T(0,0),\vect T(1,0), \vect T(1,1)$$ to each of its corners.
• Affine transformations scale areas by a factor of $$\detTwo{a_1}{b_1}{a_2}{b_2}$$ (where this value is negative when the transformation reflects orientation). This generalizes to the Jacobian $$\frac{\partial\vect T}{\partial\<u,v\>}=\detTwo {\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} {\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}}$$ for more arbitrary transformations.
• It follows that if the transformation $$\vect T(u,v)$$ transforms the region $$G$$ in the $$uv$$ plane into the region $$R$$ in the $$xy$$ plane, then $$\iint_R f(x,y)\,dA=\iint_G f(\vect T(u,v))|\frac{\partial\vect T}{\partial\<u,v\>}|\,dA$$.
• Similarly for 3D transformations $$\vect T(u,v,w)=\<x,y,z\>$$ sending $$H$$ to $$D$$, it may be shown that $$\iiint_D f(x,y,z)\,dV=\iiint_H f(\vect T(u,v,w))|\frac{\partial\vect T}{\partial\<u,v,w\>}|\,dV$$ where $$\frac{\partial\vect T}{\partial\<u,v,w\>}=\detThree {\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} {\frac{\partial x}{\partial w}} {\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}} {\frac{\partial y}{\partial w}} {\frac{\partial z}{\partial u}}{\frac{\partial z}{\partial v}} {\frac{\partial z}{\partial w}}$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 14.8

### Practice Exercises

Let the unit square have vertices $$\<0,0\>$$, $$\<1,0\>$$, $$\<1,1\>$$, and $$\<0,1\>$$. Let the unit triangle have vertices $$\<0,0\>$$, $$\<1,0\>$$, and $$\<1,1\>$$.

1. Find a transformation from either the unit square or unit triangle in the $$uv$$ plane into the given region $$R$$ in the $$xy$$ plane.
• $$R$$: the parallelogram bounded by $$y=3x+1$$, $$y=3x-3$$, $$y=x-3$$ $$y=x+1$$
• $$R$$: the triangle bounded by $$y=x$$, $$y=2x$$, $$y=6-x$$
• $$R$$: the square with vertices $$\<2,1\>$$, $$\<-2,3\>$$, $$\<0,7\>$$, $$\<4,5\>$$
• $$R$$: the triangle with vertices $$\<0,-2\>$$ $$\<-1,1\>$$, $$\<1,3\>$$
2. Evaluate the double integral with variables $$x,y$$ using the given transformation from the $$uv$$ plane.
• $$\iint_R (2y-4x)\,dA$$, $$\vect{T}(u,v)=\<u+v,2u-v+3\>$$ from the unit square into the parallelogram $$R$$ with vertices $$\<0,3\>$$, $$\<1,5\>$$, $$\<2,4\>$$, $$\<1,2\>$$.
• $$\iint_R (x+y)(x-y-2)\,dA$$, $$\vect{T}(u,v)=\<4-u-v,v-u+2\>$$ from unit triangle into the triangle $$R$$ with vertices $$\<4,2\>$$, $$\<3,1\>$$, $$\<2,2\>$$.
• $$\iint_R (x+y)e^{x^2-y^2}\,dA$$, $$\vect{T}(u,v)=\<u+2v,u-2v\>$$ from unit square into the rectangle $$R$$ bounded by $$y=x$$, $$y=x-4$$, $$y=-x$$, $$y=2-x$$.
• $$\iint_R \cos(e^x)\,dA$$, $$\vect{T}(u,v)=\<\ln (u+v+1),v\>$$ from unit triangle into the region $$R$$ bounded by $$y=0$$, $$y=e^x-2$$, $$y=\frac{e^x-1}{2}$$.

### Solutions

1. Find the transformation (solutions are not unique):
• $$\vect T(u,v)=\<2u+2v-2,2u+6v-5\>$$
• $$\vect T(u,v)=\<3u-v,3u+v\>$$
• $$\vect T(u,v)=\<4u+2v-2,-2u+4v+3\>$$
• $$\vect T(u,v)=\<u+v-1,-3u+5v+1\>$$
2. Evaluate the integral:
• $$9$$
• $$\frac{3}{2}$$
• $$\frac{1}{8}(e^8-9)$$
• $$-\frac{1}{2}\cos(3)+\cos(2)-\frac{1}{2}\cos(1)$$