# Jacobian of the transformation T:x=u, y=uv

• MHB
• karush
In summary, to graph u= 1, u= 2, uv= 1, and uv= 2 in Desmos, first replace u with x and v with y. Then, enter the functions x= 1, x= 2, y= 1/x, and y= 2/x. Finally, limit the axes to x and y values between 0 and 3 for a good graph. The typeset can be found at the given Desmos plot link.
karush
Gold Member
MHB
View attachment 8171

how do you graph this in Desmos ?

Assume the rest of the calculation is correct

Do you mean how to use Desmos to graph u= 1, u= 2, uv= 1, and uv= 2?

First, of course, Desmos graphs either y= f(x) or x= g(y) so you have to replace u with x and v with y and solve xy= 1 and xy= 2 for y (or x). that is, enter the functions x= 1, x= 2, y= 1/x, and y= 2/x. I limited the axes to that x and y were between 0 and 3 to get a good graph.

Country Boy said:
Do you mean how to use Desmos to graph u= 1, u= 2, uv= 1, and uv= 2?

First, of course, Desmos graphs either y= f(x) or x= g(y) so you have to replace u with x and v with y and solve xy= 1 and xy= 2 for y (or x). that is, enter the functions x= 1, x= 2, y= 1/x, and y= 2/x. I limited the axes to that x and y were between 0 and 3 to get a good graph.

View attachment 8176

ok here is my eventual typeset

desmos plot is here

https://www.desmos.com/calculator/cmqpsbp85y

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## 1. What is the Jacobian of a transformation?

The Jacobian of a transformation is a matrix that represents the rate of change of a transformation at a specific point. It is important in determining how a change in one set of variables affects the other set of variables in a transformation.

## 2. How is the Jacobian of a transformation calculated?

The Jacobian of a transformation is calculated by taking the partial derivatives of the transformation equations with respect to each variable, and then arranging them in a matrix. For the transformation T:x=u, y=uv, the Jacobian would be:

$\begin{bmatrix}&space;\frac{\partial&space;u}{\partial&space;x}&space;&&space;\frac{\partial&space;u}{\partial&space;y}\\&space;\frac{\partial&space;(uv)}{\partial&space;x}&space;&&space;\frac{\partial&space;(uv)}{\partial&space;y}&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;1&space;&&space;0\\&space;v&space;&&space;u&space;\end{bmatrix}$

## 3. What is the significance of the Jacobian of a transformation?

The Jacobian of a transformation is significant because it determines whether a transformation is invertible and whether it preserves the orientation of points in space. It is also used in calculating volumes and areas in multivariable calculus.

## 4. How does the Jacobian of a transformation affect the shape of a graph?

The Jacobian of a transformation affects the shape of a graph by determining how much a small change in one variable affects the other variables. It can stretch or compress the graph in different directions, depending on the values of the Jacobian matrix at a particular point.

## 5. Can the Jacobian of a transformation be negative?

Yes, the Jacobian of a transformation can be negative. This can occur when the transformation changes the orientation of points in space, such as reflecting or rotating a graph. In these cases, the Jacobian will have a negative determinant, indicating a change in orientation.

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