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Introduction to Vectors, Vector Analysis/ Calculus

Vector calculus or analysis are topics primarily invented to study electromagnetics. They are usually found in Advanced Engineering Math or Multivariable Calculus courses.

When studying vectors, you will discover abstract terms such as gradient, divergence, and curl.  Also included in the mix are partial differential equations involving separation of variables.

The study of vectors will also involve using different coordinate systems:  cartesian, cylindrical, and spherical.  The one shown below is for cartesian or rectangular coordinate systems.

A vector in the Cartesian plane, showing the p...
Image via Wikipedia
Vector in a cartesian coordinate system. a = a...
Image via Wikipedia

In vector analysis, you have vectors and scalars.

What is the difference?

Scalars consist of  just a single element.   They have magnitude but no direction.  Examples include: temperature, mass,  and length.

Vectors  contain a collection of elements like coordinates of  points (x,y,z)  found in rectangular space.  Unlike scalars, vectors have both magnitude and direction.   Other examples of vectors include: velocity, acceleration, displacement and force.  When vectors have the ample magnitude and direction, we say they are equal regardless of  their initial position.  We call the initial point the origin (tail) and arrow end the terminal point.

You can perform operations of vectors such as subtraction and addition as shown below.

The subtraction of two vectors a and b
Image via Wikipedia

You can have an array of vectors called a field:

Curl (mathematics)
Image via Wikipedia

Further discussion on vectors is described below from MIT:

Lec 1 | MIT 18.02 Multivariable Calculus, Fall 2007

Lecture 01: Dot product. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu

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