Calculus with vectors can be used to solve problems involving motion in two dimensions with variable acceleration.

bDifferentiation/b

If (\bold{r} = x\bold{i} + y\bold{j}), then:

(\bold{v} = \dfrac{d\bold{r}}{dt} = \bold{\dot{r}} = \dot{x}\bold{i} + \dot{y}\bold{j} )

(\bold{a} = \dfrac{d\bold{v}}{dt} = \dfrac{d^{2\bold{r}}{dt2}} = \bold{\ddot{r}} = \ddot{x}\bold{i} + \ddot{y}\bold{j} )

Dot notation is shorthand for differentiation with respect to time:

(\dot{x} = \dfrac{dx}{dt} ) and (\ddot{x} = \dfrac{d^2x}{dt2} )

bIntegration/b

Integration is the reverse process to differentiation, therefore:

(\bold{r} = \displaystyle\int{\bold{v}}~dt)

(\bold{v} = \displaystyle\int{\bold{a}}~dt)

The constant of integration (c) is also a vector, and should be written in the form (p\bold{i}+ q\bold{j} ).

bCalculus in kinematics for vectors/b

If (\bold{r} = x\bold{i} + y\bold{j}), then:

(\bold{v} = \dfrac{d\bold{r}}{dt} = \bold{\dot{r}} = \dot{x}\bold{i} + \dot{y}\bold{j} )

(\bold{a} = \dfrac{d\bold{v}}{dt} = \dfrac{d^{2\bold{r}}{dt2}} = \bold{\ddot{r}} = \ddot{x}\bold{i} + \ddot{y}\bold{j} )

(\bold{r} = \displaystyle\int{\bold{v}}~dt)

(\bold{v} = \displaystyle\int{\bold{a}}~dt)