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:

- Addition of vectors is commutative, and associative.
- 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:

Position vector | Place the tail of a vector at the origin, and then locate a point in space.`r = x i + y j + z k` . |

Displacement vector | The difference between 2 position vectors.`r` |

## 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 = a`_{1} i + a_{2} j + a_{3} k
B = b_{1} i + b_{2} j + b_{3} k
A ยท B = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

## 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 ร 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 = r`_{0} + 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 vector identities | Left hand side is scalar. Right hand side is scalar. |

Vector vector identities | Left hand side is vector. Right hand side is vector. |

## 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) = u_{1}(x, y, z, t) **i** + u_{2}(x, y, z, t) **j** + u_{3}(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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