Calculus of vectors

Differentiation

Vector-valued functions of scalars

A vector-valued function of a scalar (a real mouthful to say) is a function which outputs a vector value and is a function of a scalar variable. Given a function, \mathbf{a}(x), the derivative of \mathbf{a}(x) is similar to the derivative of a scalar valued function of a scalar. $$ \frac{d\mathbf{a}}{dx} = \lim_{\Delta x\to 0} \frac{\mathbf{a}(x + \Delta x) - \mathbf{a}(x)}{\Delta x} $$ The rules for calculating the derivatives of such functions draw parallels with the rules for calculating the derivatives of scalar-valued functions. $$ \frac{d}{dx}(h \cdot g) = \frac{dh}{dx} \cdot g + h \cdot \frac{dg}{dx} $$

Vector-valued vector functions (Jacobians)

Integration

Line integrals

Surface integrals

Volume integrals

Vector operators