The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus. Vector calculus is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. The content in this Textmap is complemented by Vector Calculus Modules in the Core and the Vector Calculus (UCD Mat 21D) Libretext.
- In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b,c) . Since Euclidean space is 3-dimensional, the graph of f consists of the points (x,y,z)=(x,y,f(x,y)).
- In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions, that is, for each input value we get a position in space. Now we turn to functions of several variables, meaning several input variables, functions.
- The multiple integral is a generalization of the definite integral with one variable to functions of more than one real variable. For definite multiple integrals, each variable can have different limits of integration.
- A line integral is an integral where the function to be integrated is evaluated along a curve and a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). Surface integrals have applications in physics.