Calculus · Quick Reference

Vector Calculus Cheat Sheet

Multivariable Calculus · Lesson 17/20

In one line: vector calculus studies functions that output vectors (called vector fields) and the operations — gradient, divergence, curl — that describe how they behave in space.

Key Ideas

1Vector Fields. A vector field assigns a vector to every point in space, such as F(x,y) = (P(x,y), Q(x,y)) — think of wind speed and direction at every point on a map.

2Scalar vs. Vector Functions. A scalar function outputs a single number (like temperature); a vector function outputs a vector (like velocity).

3Line Integrals (Concept). A line integral adds up a scalar or vector field's values along a curve, rather than over an interval on the x-axis.

4Key Vector Operators. The three central tools are the gradient (turns a scalar field into a vector field), divergence (turns a vector field into a scalar), and curl (turns a vector field into another vector field, in 3D).

5Why It Matters. Vector calculus underlies physics topics like fluid flow, electromagnetism, and gravitational fields.

Worked Examples

Given the vector field F(x,y) = (y, -x), find F at the point (2,3).

F(2,3) = (3, -2)

Is g(x,y) = x^2 + y^2 a scalar field or a vector field?

It is a scalar field

Is F(x,y) = (2x, 3y) a scalar field or a vector field?

It is a vector field

Practice Yourself

Given F(x,y) = (x, y), find F at (1,1).

(1,1)

Is f(x,y,z) = x+y+z a scalar or vector field?

Scalar field

Is F(x,y) = (y^2, x^2) a scalar or vector field?

Vector field

Given F(x,y) = (2y, -2x), find F at (0,3).

(6, 0)