4. Change of Variables Flashcards
What does “Change of variables” refer to, in n-dimensions?
“Change of variables” is just a generalization of u-substitution to n-dimensions. In u-substitution, we use a function u(x) to replace x with u, converting the integral to an easier one. In change of variables in n-dimensions, we use a function g(x1, x2, …., xn) to replace a vector x with a different one, converting the integral to an easier one.
What does this notation mean?
The partial derivative of the jth component of g with respect to the ith component of x.
Explain in words what the Jacobian matrix of a function is?
The Jacobian matrix is made up of partial derivatives of the function; in each row a different component of the function gj is being derived; in each column the partial derivative is taken with respect to a different variable xi
How is the Jacobian matrix written and defined (symbolically)?
Explain, in loose terms, why the absolute value of the determinant of the jacobian matrix is in the change of variables formula.
The determinant of the jacobian is there because of the chain rule (in u-substitution, you have to substitute the differential of integration dx with a du term that represents the chain rule). The absolute value is just so that we get the true volume, not the orientation-dependent signed volume.
What happens to the determinant of a matrix if you switch two columns or two rows?
The determinant changes sign.
How can we think of the determinant as a function?
The determinant can be thought of as a mapping of Rnxn to R.
What happens to the determinant of a matrix when you take the transpose of the matrix?
Nothing (since you are allowed to expand along any row or column when computing the determinant)
Can you multiply determinants together?
Yes, det[AB] = det[A]det[B]
What does it mean for the determinant of a matrix to be zero?
It means the matrix is not invertible, or that the vectors of the matrix are not linearly independent.
What hapens to the determinant of a matrix if you multiply one of the rows or columns by a scalar?
The determinant is multiplied by that scalar (think of expansion along that row or column)
How do you make the determinant of a matrix as easy as possible to compute?
Expand along the row or column with the greatest number of zeros
Describe what the change of variable formula can be useful for, and what is the most difficult part of using it?
If we have a really tough region of integration, we can transform the region of integration into something simple over which we know how to integrate. For example, we can integrate over any general triangle by transforming it into a unit triangle via the change of variable formula. The most difficult part of performing a change of variables is figuring out the transformation function.
In general, of what form is the mapping in the change of variable formula?
A linear transformation [A]x + b.
Such a mapping is called an “Affine map”
In general, how do you compute the “affine map” from one region to another?
Solve for the matrix [A] of the affine map by plugging in the vertices of the first region into the affine map equation [A]x+b = y (where y is the coordinate in the new region) and solving for the columns of A and the vector b.
What is special about Affine maps?
The Jacobian matrix of an Affine map is simply the Affine map’s transformation matrix [A], meaning Affine maps are particularly convenient for changing of variables.
If two regions of integration are related by an affine map [A]x+b, what can we say about the volumes of the regions?
They will be related by the determinant of the jacobian, which is just the determinant of the matrix [A]:
voln(x) = det[A]voln(y)
What shape is described by this set, and how do you know?
It is a shape that has an ellipsical trace in the xy-plane. This can be seen by imagining z to be a constant, in which case the set would appear to describe a disk that has been morphed by two scaling factors (which is what an ellipse is).
Express this x,y,z-space region in r,theta,z-space
When performing a change of variables, don’t forget to_______
Rewrite the bounds of integration in terms of the new variables (this was the whole reason you even changed variables, remember)
What can we say about the determinant of a triangular matrix?
It is equal to the product of the diagonal entries
What does it mean to say that the determinant is linear with respect to each vector inside a matrix?
