# 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 gradientf 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
• du|⟨dx,dy,dz⟩=v = ∇f(x, y, z) ⋅ v

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