BitWithBite
BitWithBite / Calculus / Vector Calculus
📘 Lesson 17 of 20 · Calculus
Multivariable Calculus

➡️ Vector Calculus

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

Course progress: 85%

01 Key Concepts

Vector 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.

Scalar vs. Vector Functions

A scalar function outputs a single number (like temperature); a vector function outputs a vector (like velocity).

Line 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.

Key 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).

Why It Matters

Vector calculus underlies physics topics like fluid flow, electromagnetism, and gravitational fields.

02 Solved Examples

Example 1 Given the vector field F(x,y) = (y, -x), find F at the point (2,3).
  1. Substitute x=2, y=3 into F(x,y) = (y, -x).
  2. F(2,3) = (3, -2).
Answer: F(2,3) = (3, -2)
Example 2 Is g(x,y) = x^2 + y^2 a scalar field or a vector field?
  1. g(x,y) outputs a single real number for every point (x,y), not a pair or triple of numbers.
Answer: It is a scalar field
Example 3 Is F(x,y) = (2x, 3y) a scalar field or a vector field?
  1. F(x,y) outputs a pair of numbers (a vector) for each point (x,y).
Answer: It is a vector field

03 Practice Questions

1Given F(x,y) = (x, y), find F at (1,1).
(1,1)
2Is f(x,y,z) = x+y+z a scalar or vector field?
Scalar field
3Is F(x,y) = (y^2, x^2) a scalar or vector field?
Vector field
4Given F(x,y) = (2y, -2x), find F at (0,3).
(6, 0)
5What does a vector field assign to each point in space?
A vector (magnitude and direction)

📄 Vector Calculus — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet