PX153 - G2 - the total differential, and exact and inexact differentials

if we think of a 3D plot of $f (x, y)$ , it is clear that the gradient changes depending on the direction
we define the total differential as:

d f = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y

more generally, we can write the small change as you change variables by a small amount as:

d f = A (x, y) d x + B (x, y) d y

- this is no longer explicitly l\inked to a function, $f (x, y)$

we say that a differential is exact if and only if it can be integrated to a function of the variables, $f (x, y)$ .
an exact differential can always be integrated to a function, and the differential found from the derivative of a function is always exact
comparing the two equations above gives:

A (x, y) = \frac{\partial f}{\partial x} a n d B (x, y) = \frac{\partial f}{\partial y} $ $ s u c h t h a t : $ $ \frac{\partial A}{\partial y} = \frac{\partial^{2} f}{\partial y \partial x} = \frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial B}{\partial x}

so, a differential is exact if and only if:

\frac{\partial A}{\partial y} = \frac{\partial B}{\partial x}

eg: is $x . d y + 2 y . d x$ exact?
- compare with $d f = A . d x + B . d y$
- $A = 2 y$ and $B = x$
- $\frac{\partial A}{\partial y} = 2 \neq \frac{\partial B}{\partial y x} = 1$
- so, not an exact differential
we can extend this to multiple variables:

d f = \sum_{i = 1}^{n} g_{i} (x_{1} . . . x_{n}) d x_{i}

is exact if

\frac{\partial g_{i}}{\partial x_{j}} = \frac{\partial g_{j}}{\partial x_{i}} $ $ f o r a l l p a i r s o f $ i $ a n d $ j $ - e g : s h o w t h a t t h e f o l l o w i n g p a i r s o f d i f f e r e n t i a l i s e x a c t a n d f i n d t h e f u n c t i o n f r o m w h i c h t h e d i f f e r e n t i a l i s d e r i v e d : $ $ (y + z) . d x + x . d y + x . d z

d f = g_{x} . d x + g_{y} . d y + g_{z} . d z

$g_{x} = y + z$ , $g_{y} = x$ , and $g_{z} = x$
$\frac{\partial g_{x}}{\partial y} = 1 = \frac{\partial g_{y}}{\partial x}$ , $\frac{\partial g_{x}}{\partial z} = 1 = \frac{\partial g_{z}}{\partial x}$ , $\frac{\partial g_{y}}{\partial x} = 0 = \frac{\partial g_{z}}{\partial y}$
- so, it is exact:
$$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz$$
$$\frac{\partial f}{\partial x} = y+z \implies f(x,y,z) = x(y+z)+h_{x}(y,z)+c$$
$$\frac{df}{dy}= x \implies f(x,y,z) = xy + h_{y}(x,z) +c$$
$$\frac{df}{dz}= x \implies f(x,y,z) = xz + h_{z}(x,y) +c$$
- these are satisfied by: $f (x, y, z) = x (y + z) + c$

- the constant of integration could involve other variables held constant