# Vector Calculus: Basics Vector Calculus is also known as Multivariate Calculus or “Calculus 3”. Calculus 1 and 2 are Differential Calculus and Integral Calculus respectively (both are single variate).

## Vectors

A vector is a quantity that has a length associated with it, and a direction. It is not anchored in any particular spot. Besides, scalars are quantities that have only length, they are just numbers. Vectors follow a few rules:

1. Addition of vectors is commutative, and associative.
2. Multiplication of vectors by a scalar is distributive. and will not change the direction of the vectors, but just their length.

## Cartesian Coordinates

In Cartesian coordinate system, you have 3 dimensions: x, y, z. There are 2 kind of vectors widely used:

## Dot Product

Dot product is also called scalar product, because the multiplication of 2 vectors results in a scalar. You are given vectors and you are free to choose any coordinate system you want.

``````A = a1 i + a2 j + a3 k
B = b1 i + b2 j + b3 k
A · B = a1 b1 + a2 b2 + a3 b3 ``````

## Cross Product

Cross product is also called vector product. The way to remember what the vector is, is to use a three-by-three determinant. Use right hand rule to determine the direction of the product: suppose A × B, point your fingers in the direction of A, curl your fingers to the vector B, then the thumb is pointing to the direction of the product.

## Analytic Geometry

We may use 2 points (position vectors) to define the parametric equation of a line, where r and r0 are both position vectors and u is velocity vector:

``r = r0 + u t``

3 points (position vectors) could define a plane. We could get 2 displacement vectors from the 3 position vectors. Then the cross product of the 2 displacement vectors are perpendicular the the plane. The dot product of this perpendicular vector to the plane and any vector that lies in the plane is zero.

## Vector Identities

Kronecker-Delta, Levi-Civita symbol and Einstein Summation Conventions are used to prove some vector identities. There are 2 types of vector identities:

## Scalar and Vector Fields

We are interested in the phenomenon that vary in space, and vary in time, and we represent those phenomenon using the concept of fields. A position vector r is used to locate where you are in the space. A variable t is used to locate where you are in time.

Then we could define a scalar field, for example: temperature:

``T(r, t) = T(x, y, z, t)``

Or a vector field, for example velocity:

``u(r, t) = u1(x, y, z, t) i + u2(x, y, z, t) j + u3(x, y, z, t) k``

These fields are usually found through a governing equation, which are called partial differential equations.

## My Certificate

For more on Vector Calculus: Basics, please refer to the wonderful course here https://www.coursera.org/learn/vector-calculus-engineers

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