Multivariable Calculus

In this unit we've covered a lot of different function types. Be sure you can recognize each one both from its functional form and its name. The following list provides a summary of the unit.

• \(y=f(x)\) or \(f\colon \mathbb{R}\to\mathbb{R}\) (functions of a single variable)

• \(\vec r(t)=(x,y)\) or \(f\colon \mathbb{R}\to\mathbb{R}^2\) (parametric curves)

• \(\vec r(t)=(x,y,z)\) or \(f\colon \mathbb{R}\to\mathbb{R}^3\) (space curves)

• \(\vec r(u,v)=(x,y,z)\) or \(f\colon \mathbb{R}^2\to\mathbb{R}^3\) (parametric surfaces)

• \(z=f(x,y)\) or \(f\colon \mathbb{R}^2\to\mathbb{R}\) (functions of two variables)

• \(z=f(x,y,z)\) or \(f\colon \mathbb{R}^3\to\mathbb{R}\) (functions of three variables)

• \(\vec T(u,v)=(x,y)\) or \(f\colon \mathbb{R}^2\to\mathbb{R}^2\) (2D transformation)

• \(\vec T(u,v,w)=(x,y,z)\) or \(f\colon \mathbb{R}^3\to\mathbb{R}^3\) (3D transformation)

• \(\vec F(x,y)=(M,N)\) or \(f\colon \mathbb{R}^2\to\mathbb{R}^2\) (vector fields in the plane)

• \(\vec F(x,y,z)=(M,N,P)\) or \(f\colon \mathbb{R}^3\to\mathbb{R}^3\) (vector fields in space)