Formula: Gradient of a scalar function

Formula: Gradient of a scalar function
Vector plot of the gradient of the scalar function x²+5xy

Gradient field

Applying the Nabla operator to a scalar field \(f\) results in a vector field (gradient field) with three components. At a point \((x,y,z)\) the vector \(\nabla \, f(x,y,z)\) points in the direction of the largest increase of \(f\).

Here \(\nabla\) is the Nabla operator. This is a vector operator with which vectorial derivatives like gradient, divergence or rotation can be formed.

Scalar function

A function depending on three coordinates \(x\), \(y\) and \(z\), which must be differentiable. For example: \( f(x,y,z) = x^2 + 5yz + z \).

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