Vector Calculus

Vector calculus, also known as vector analysis, is a branch of mathematics that deals with vector fields, which are mathematical objects that assign a vector to every point in a region of space.

Vector Fields

A vector field in two or three dimensions is a function that assigns a vector to every point in a plane or space. Vector fields can represent a variety of physical quantities, such as velocity or force fields.

Gradient

The gradient of a scalar field (a function that assigns a scalar to every point in space) is a vector field that points in the direction of the greatest rate of increase of the scalar field. It is denoted by the symbol ∇ (called "nabla").

Divergence

The divergence of a vector field is a scalar function that provides a measure of the quantity of the vector field that is originating from a given point. It's the dot product of the nabla operator and the vector field.

Curl

The curl of a vector field is a vector function that describes the rotation of the field. It's the cross product of the nabla operator and the vector field.

Line Integrals, Surface Integrals, and Volume Integrals

Line integrals, surface integrals, and volume integrals are techniques for integrating functions along a line, over a surface, and throughout a volume, respectively. These techniques are useful for calculating quantities such as work done by a force field along a path, or the flux of a vector field across a surface.

Fundamental Theorems

There are several key theorems in vector calculus that relate the different operations and types of integrals:

  • Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C.

  • Stokes' Theorem relates a surface integral of a curl over a surface S to a line integral of the vector field around the boundary curve of S.

  • Divergence Theorem (also known as Gauss's theorem) relates the flux of a vector field across a closed surface to the triple integral of the divergence of the field throughout the volume enclosed by the surface.

These theorems are fundamental to the study of vector calculus and have widespread applications in physics and engineering.