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Calculus/Partial Differentiation

1.

If f(x,y) is a function of two variables, x and y, then

$\begin{displaymath}\left ( \frac{\partial f(x,y)}{\partial y} \right )_x = \lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h} \end{displaymath}$

(a)

If $p=\frac{nRT}{V}$ then

$\begin{displaymath}\left ( \frac{\partial p}{\partial V} \right )_{T,n} = -\frac{nRT}{V^2} \end{displaymath}$

(b)

If $p=\frac{nRT}{V}$ then

$\begin{displaymath}\left ( \frac{\partial p}{\partial T} \right )_{V,n} = \frac{nR}{V} \end{displaymath}$

(c)

The double differentiation is order independent, i.e.,

$\begin{displaymath}\left ( \frac{\partial \left ( \frac{\partial p}{\partial T} ... ...=\frac{\partial^2p}{\partial T \partial V} = -\frac{nR}{V^2} \end{displaymath}$

If f(x,y) is a function of two variables, x and y, then

$\begin{displaymath}\left ( \frac{\partial f(x,y)}{\partial y} \right )_x = \lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h} \end{displaymath}$

1.

If $p=\frac{nRT}{V}$ then

$\begin{displaymath}\left ( \frac{\partial p}{\partial V} \right )_{T,n} = -\frac{nRT}{V^2} \end{displaymath}$

2.

If $p=\frac{nRT}{V}$ then

$\begin{displaymath}\left ( \frac{\partial p}{\partial T} \right )_{V,n} = \frac{nR}{V} \end{displaymath}$

3.

The double differentiation is order independent, i.e.,

$\begin{displaymath}\left ( \frac{\partial \left ( \frac{\partial p}{\partial T} ... ...=\frac{\partial^2p}{\partial T \partial V} = -\frac{nR}{V^2} \end{displaymath}$

2002-06-14