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.
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