divergence

• Input: a vector field.
• Output: Take the derivative of along each component. Sum all derivatives.
• For a scalar field, the gradient produces a vector field. Taking the divergence of this vector field gives the laplacian
• On a manifold, some extra consideration needs to be made, because if you have an \(m < n\) dimensional manifold embedded in \(R^n\), you can't take partial derivatives along all \(n\) axis in the usual way, since you're restricted to move along the manifold. See justin solomon lecture for details.