Intuitive

Concrete

• A Survival Guide to Vector Calculus by Aylmer Johnson

Recommended Textbooks:

• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard - Extremely student friendly, lots of margin notes that talk about the "soft" stuff that's so crucial to the actual practice of math. Reading just the margins jumps your mathematical maturity by 2 years.
• 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

Geometric Calculus:

• Geometric Calculus R & D website

Geometric Calculus videos:

• Geometric Calculus 0 Alan Macdonald
• Tutorial on Geometric Calculus David Hestenes

Abstract

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

History

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