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
- du =
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)⟩
- du =
∇f(x, y, z) ⋅
⟨dx, dy, dz⟩
- ιv du =
v ⋅
∇f(x, y, z)
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This web page was written between in 2014 by Toby Bartels,
last edited on 2014 May 4.
Toby reserves no legal rights to it.
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http://tobybartels.name/MATH-2080/2014SP/gradient/.
