Matrix determinants and differentiation

In summary, The conversation discusses using implicit differentiation to find the partial derivatives of a system of equations. The method used involves using Cramer's rule to solve the system, which is not as efficient as Gaussian elimination but easier to keep track of the numerator and denominator of the solutions. The conversation also includes a discussion about formatting mathematical equations using LaTeX.
  • #1
tickle_monste
69
1
I'm having trouble understanding where this concept comes from:
Step 1) If you start out with the following two equations

v + log u = xy
u + log v = x - y.

Step 2) And then perform implicit differentiation, taking v and u to be dependent upon both x and y:
(d will represent the partial derivative symbol):

dv/dx + (1/u)du/dx = y
du/dx + (1/v)dv/dx = 1

I can do some simple Gaussian reduction and obtain:
du/dx = [u(v-y)]/[uv-1] which is the same answer my book gives, the only difference is that my book uses this method to find du/dx:

Step 3)
det(Row 1:yu, u; Row 2: v, 1)/det(Row 1: 1, u; Row 2: v, 1)
which reduces to: (yu)(1) - (u)(v))/(1)(1) - (u)(v).
If the two matrices were A, and B, respectively, then:
a11 = yu, a12 = u, a21 = v, a22 = 1.
b11 = 1, b12 = u, b21 = v, b22 = 1
(sorry, I don't know how to put a real matrix into here)

And my problem is that I just don't understand where these matrices came from. I think this may be some formula from linear algebra that I just don't remember, but my book gives no reference to what it's doing, and goes directly from step 2 to step 3, so I'm really kind of lost right now. Any help would be appreciated.
 
Physics news on Phys.org
  • #2
It looks they used Cramer's rule to solve the linear system of equations, rather than Gaussian elimination. (They also cleared denominators before solving)

I couldn't guess why -- Cramer's rule is rather good for proving theorems but rather bad for calculation. I suppose the 2x2 case isn't quite so bad, though.
 
  • #3
t_m:
Your system of equations for the derivatives is
[tex]
\begin{align*}
\frac 1 u \frac{du}{dx} + \frac{dv}{dx} & = y \\
\frac{du}{dx} + \frac 1 v \frac{dv}{dx} & = 1
\end{align*}
[/tex]

Multiply the equations to clear fractions:

[tex]
\begin{align*}
\frac{du}{dx} + u \cdot \frac{dv}{dx} & = uy\\
v \cdot \frac{du}{dx} + \frac{dv}{dx} & = v
\end{align*}
[/tex]

This can be put into matrix form as

[tex]
\begin{bmatrix}
1 & u \\ v & 1
\end{bmatrix}
\,
\begin{bmatrix}
{du}/{dx} \\ {dv}/{dx}
\end{bmatrix} =
\begin{bmatrix}
uy \\ v
\end{bmatrix}
[/tex]

and, as Hurkyl said, Cramer's rule was apparently used. My only reasoning about why it was used in this case: by using Cramer's rule the numerator and denominator of the solutions are easier to keep track of than they are when you use Gaussian elimination.
 
  • #4
I see, now it makes sense. Thanks for the help! Also, when I hover the mouse over those matrices you used it has some LaTeX script, can I just write that script into the text box on here? Or do you have do something else?
 
  • #5
you need to enclose the latex markup in delimiters, like this:

[ t e x ]
your markup goes in here
[/ t e x]

I left the spaces in "tex" and "/tex" to avoid problems with my note. You do need the [] pair in each case.
 
  • #6
Hurkyl said:
I couldn't guess why -- Cramer's rule is rather good for proving theorems but rather bad for calculation. I suppose the 2x2 case isn't quite so bad, though.

For two by two it doesn't matter but I thought Cramer's rule was good for symbolic calculations but not so good for numeric calculations.
 

What is a determinant?

A determinant is a mathematical value that can be calculated from a square matrix. It represents the scaling factor of the linear transformation described by the matrix.

How do you find the determinant of a matrix?

To find the determinant of a matrix, you can use the method of cofactor expansion or the rule of Sarrus for 2x2 matrices. The determinant can also be calculated using row operations, such as Gaussian elimination.

What is the relationship between matrix determinants and linear transformations?

The determinant of a matrix represents the scaling factor of the linear transformation described by the matrix. This means that a larger determinant indicates a larger scaling factor and a more significant change in the transformation.

What is differentiation?

Differentiation is a mathematical operation that calculates the rate of change of a function with respect to its independent variable. It is used to find the slope of a curve at a specific point and is an essential tool in calculus.

How are matrix determinants and differentiation related?

Matrix determinants and differentiation are related in that both concepts involve calculating the rate of change. In the case of matrix determinants, it is the scaling factor of a linear transformation, and in differentiation, it is the slope of a curve. Additionally, the derivative of a matrix is known as the Jacobian matrix, which is closely related to matrix determinants.

Similar threads

Replies
2
Views
830
Replies
22
Views
2K
Replies
1
Views
2K
Replies
19
Views
2K
  • Calculus and Beyond Homework Help
Replies
19
Views
674
  • Calculus
Replies
2
Views
1K
Replies
4
Views
188
Replies
3
Views
215
Back
Top