[Mikez]

For heat capacity at a constant volume I usually see it's

(triangle) U / (triangle T)

is the triangle delta the same as having a d infront?? dU/dT and similarly is this the same as having the partial derivate symbol of ∂.U/ ∂T?

When would I use the different type of notation?

Thank you

[Juan R.]

This is a purely mathematical question. Open a textbook and learn the difference between a ratio  (:delta: f / :delta: x) a derivative (df/dx) and a partial derivative (∂f/∂x)

[Mikez]

this is not in my book, I only need to know the difference between

delta, d, and partial derivative

and what happens when I take the integral of those

that is all the calculus I need so I do the derivations for my class

[rabolisk]

It's hard to teach calculus over the net. I would go over to wolfram and learn the differences...

In general, though, delta simply means change. As far as partial derivatives go, they are used because most functions (or dependent variables) depend on more than one variable. So a partial derivative of a function f with respect to variable x signifies the (infinitesimal) change in f over the (infinitesimal) change in x, while holding all the other variables (y, z, etc..) constant.

[Juan R.]

this is not in my book, I only need to know the difference between

delta, d, and partial derivative

and what happens when I take the integral of those

that is all the calculus I need so I do the derivations for my class

As said before, this is a purely mathematical question which is explained in textbooks of mathematics; not a surprise that is not explained in your book.

 :delta: denotes a variation and (d/dx) is an operator of differentiation (sometimes denoted as D in math textbooks) defined as the limit of the ratio of the variations of the function and of the variable when the variation of variable goes to zero. (∂/∂x) is the natural generalization of this operator to multivariable functions. Integration can be understood as the inverse operator of the differentiation operator I = D^-1.

For instance for f=f(x) if D(f)=g then the integral of g is I(g)=f+cte because D(I(g))=DD^-1(g)=D(f)+D(cte)=g+0

[Mikez]

thanks