Reconstructing%20a%20Function%20from%20its%20Gradient

1
Reconstructing a Function from its Gradient
2
Differentials
We begin by reviewing the one-variable case. If f
is differentiable at x, then for small h, the
increment ?f f (x h) - f (x) can be
approximated by the differential d f f(x) h.
3
Differentials
4
Differentials
is called the increment of f , and the dot product
is called the differential (more formally, the
total differential).
5
Differentials
As in the one-variable case, for small h, the
differential and the increment are approximately
equal
6
Differentials
7
Differentials
8
Vector Form of Differentials
9
Example
10
Example
11
Examples
12
Vector Functions
13
Example
14
Exact Differentials
15
Examples
16
Example
17
Theorem
18
Examples
19
Examples