Gradient ( $\nabla f$ )

The gradient of a scalar-valued differentiable function is the vector of its first partial derivatives.

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- Gradient field Gradient

Gradient field

A gradient field is a vector field of the form

\nabla f

for a scalar potential

f

. Its curl vanishes wherever the required second derivatives commute.

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Ciro Santilli  40 Updated 2025-07-16

Takes a scalar field as input and produces a vector field.

Mnemonic: the gradient shows the direction in which the function increases fastest.

Think of a color gradient going from white to black from left to right.

Therefore, it has to:

take a scalar field as input. Otherwise, how do you decide which vector is larger than the other?
output a vector field that contains the direction in which the scalar increases fastest locally at each point. This has to give out vectors, since we are talking about directions

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