Math Is Fun Forum

I'm saying that if both matrices aren't invertible, then there are cases where ab≠ba. I am not saying it's always the case if at least matrix is singular, because it isn't case (any matrix, singular or not, commutes with the identity matrix).

There is no way to do it, because it isn't true.

Except the inverses might not exist.

How much does it differ from Texas?

Okay, that is true.

Most of them will be unnecessary, but there are certain types of sentences that are hard to say without double negation. For example "I've never been there" would require double negation.

Bengali grammar? In Serbian, double negations do not cancel out!

Hi Agnishom

What's happening is you need to do this test on two functions. You take the derivative of one w.r.t. x and the other w.r.t. y and see if they're equal. If they are, you can conclude they are the y and x (in that order) derivatives of the same function. Why does it work? Because of the identity you posted! Every functions second order mixed partial derivative is unique, no matter which variable you derive with respect to first. Thus, given two functions p(x,y) and q(x,y), if there is a function f(x,y) such that d/dx f(x,y)=p(x,y) and d/dy f(x,y)=q(x,y), then we know that this equation holds:
d/dy d/dx f(x,y)=d/dx d/dy f(x,y) which is equivalent to:
d/dy p(x,y)=d/dx q(x,y)

As you can see, this is an obvious statement and is not of much use. But what the Euler's criterion states is the converse of the above i.e. "if two functions, p(x,y) and q(x,y), satisfy the last equation I wrote above, then there is a function f(x,y), such that d/dx f(x,y)=p(x,y) and d/dy f(x,y)=q(x,y).

Where did you come across the? It might help us know how to approach it.

The 6 and 4 are for choosing the suits.

There are 13*12*4*6 full houses.

So, how does one begin to get into p?