Why is it interesting?

Vector calculus is an important tool, whenever we want to understand systems where directions play a role. A vector is an arrow that points in some direction. Thus, a vector is a tool to denote a direction.

This is needed, for example, to describe in which spatial direction a ball moves or how a fluid flows.

A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction. In keeping with this definition, speed (how fast an object is going) is not represented by a vector, but velocity (how fast and in which direction an object is going) does qualify as a vector quantity. Another example of a vector quantity is force, which describes how strongly and in what direction something is being pushed or pulled. But temperature, which has magnitude but no direction, is not a vector quantityA Student's Guide to Vectors and Tensors by Daniel A. Fleisch

Important Vector Calculus Concepts:

• Curl
• Gradient
• Flux
• Divergence
• Dot Product / Scalar Product
• Cross Product
• Gauss' Divergence Theorem
• Stokes's Theorem

Layman

Student

• A Survival Guide to Vector Calculus by Aylmer Johnson

Recommended Textbooks:

• http://mathinsight.org/thread/vector_algebra
• A nice introduction can be found in Section 3 of Vol. 2 of Feynman's Lectures on Physics, which are available here
• A Student's Guide to Vectors and Tensors by Daniel A. Fleisch
• DIV, Grad, Curl, and All That: An Informal Text on Vector Calculus by H. M Schey
• Mathematical Tools for Physics by James Nearing

Researcher

Examples

Example1

Example2:

FAQ

History

https://en.m.wikipedia.org/wiki/A_History_of_Vector_Analysis