Some notes about Vector Calculus. Mainly Divergence, Gradient and Curl.

Nabla Operator ($\nabla$)

For a 3-dimensional cartesian coordinate system:

Sometimes $\vec\nabla$ is written instead of $\nabla$ to show that it’s basically a vector.

The Nabla Operator can be seen as “Vector of partial derivative operators”. The dimension is same as the input dimension.

Operator: takes as input a function an outputs an other function.

Gradient ($\operatorname{grad}$)

The gradient represents the slope of the tangent of the graph of a function.

Input: A Scalar field

Output: A Vector field

The direction of the gradient at one position shows the direction of the biggest change (steepest ascent) in the scalar field. The vectors are pointing “uphills”. The length (absolute value) of the gradient at that point is a measurement of the change (slope, steepness).

$e_i$: The orthogonal unit vectors pointing in the coordinate directions

The Gradient is always perpendicular to the Contour lines.

Divergence ($\operatorname{div}$)

The divergence is a vector operator that measures the magnitude of a vector field’s source or sink. It is a scalar field with signed components (positive values for sources, negative values for sinks).

Input: A Vector field

Output: A Scalar field

The divergence is the scalar product between the Nabla operator $\nabla=(\tfrac{\partial}{\partial x_1},\ldots,\tfrac{\partial}{\partial x_n})$ and the vector $\vec F = (F_1, \ldots,F_n)$.