Differentials, gradients, and partial derivatives
Given u = f(x, y, z),
we may speak of the differential du of this quantity
as well as the gradient ∇f of the function.
Within du,
we can find the partial derivatives of u
with respect to x, y, and z
(holding the other two fixed);
within ∇f,
we can find the partial derivatives of f
with respect its first, second, or third arguments.
Here are a bunch of relationships between these.
- du =
(∂u/∂x)y,z dx +
(∂u/∂y)x,z dy +
(∂u/∂z)x,y dz
- df(x, y, z) =
D1f(x, y, z) dx +
D2f(x, y, z) dy +
D3f(x, y, z) dz
- D1f(x, y, z) =
(∂u/∂x)y,z
- D2f(x, y, z) =
(∂u/∂y)x,z
- D3f(x, y, z) =
(∂u/∂z)x,y
- ∇f(x, y, z) =
⟨D1f(x, y, z),
D2f(x, y, z),
D3f(x, y, z)⟩
- df(x, y, z) =
∇f(x, y, z) ⋅
⟨dx, dy, dz⟩
- ⟨du | v⟩ =
∇f(x, y, z) ⋅
v
Go back to the course homepage.
This web page was written between in 2014 by Toby Bartels,
last edited on 2014 January 29.
Toby reserves no legal rights to it.
The permanent URI of this web page
is
http://tobybartels.name/MATH-2080/2014WN/gradient/.
