Overview Triple integrals, surface integrals and contour integrals, and bridges between them.

Stokes’ for work (in space) The work done by a vector field along a closed curve, can be replaced with a double integral.

Curl (in space) The curl measures the value of the vector field to be conservative. For a velocity field, curl measures the rotation component of the motion.

Gradient Field (in space) Test for gradient field. Find the potential of a gradient field.

Diffusion/heat (in space) The Diffusion equation governs motion of e.g. smoke in unmovable air, or dye in a solution.

Divergence (in space) Divergence measures how much the flow is expanding. It singles out the stretching component of motion.

Work (in space) Whenever a force is applied to an object, causing the object to move, work is done by the force.

Vector Fields; Surface integrals; Flux (in space) About vectors in space and determining the surface vector, using flux as an example.

Triple Integrals Using triple integrals we can find volume between two surfaces.

Flux; Green’s (in plane) Flux is the amount of something (water, wind, electric field, magnetic field) passing through a surface.

Curl; Green’s (in plane) For a velocity field, curl measures the

*rotation*component of the motion. Curl also measures how far the vector field is from being conservative. Matrices Matrices can be used to express linear relations between variables. For example when we change coordinate systems.

Vectors Vectors do not have a start point, but do have a magnitude (length) and direction. They are described in terms of the unit vectors, or using angle brackets notation.

Gradient Field (in plane) When a vector field is a gradient of function f(x,y), it is called a gradient field.

Vector fields; Line Integrals; Work (in plane) Line integrals in scalar field; line integrals in vector field;.

Double Integrals Find the volume between a function f(x,y) and a certain region in the xy-plane.

Foundations Function graphs; parametric curves.

Overview Vector calculus is about differentiation and integration of vector fields. This article gives an overview of the differentiation. operations.