Multivariable Calculus

In calculus, we have dealt
with functions
of x in two dimensional space. Multivariable Calculus, also known as Vector Calculus,
deals with functions of two variables in 3 dimensional space, as well as computing
with vectors instead of lines.

In single variable calculus, we see that y is a function of x

In multivariable calculus, z is a function of both x and y

Multivariable calculus extends the notion of
in three dimensional space. New concepts such as partial
derivatives, directional derivatives, line integration and multiple integrals are

When plotting in 3 dimensional space, there needs to be new notation for various
concepts that will be introduced. Multivariable calculus also introduces vectors,
which may have been introduced in physics. Calculus dealing with vectors will take
into consideration not only magnitude but direction as well.

Multivariable calculus is a broad field that applies to physics and other sciences
more so than single variable calculus. Many of the theorems provided in vector calculus
are essential for solving problems in physics, which are mostly multidimensional.
Multivariable calculus studies the rates of change and works in more than two dimensions.
In single variable calculus, an object might move through the air and we can calculate
its velocity at any given point. In vector calculus, we can then take into account
air resistance and the force of gravity acting on the object.

It is important to have a thorough understanding of the vectors and what they are
used for before moving on to differentiation and integration in multivariable calculus.
Since it is such a broad topic, we will start with an introduction to vectors in
two and three dimensions as well as plotting lines and planes. Let’s start with
an introduction to vectors in two dimensional space.

Introduction to Vectors

Main Lesson:

An introduction to vectors in two dimensional space.

Key topics include Scalars and Vectors, Vector Notation, Vector Equalities and Operations,
the Dot Product, the Angle between two vectors, and the Projection of a vector.

Vectors in Three Dimensional Space

Main Lesson:
Vectors in 3 Dimensional Space

An introduction to working with vectors in three dimensional space and working with
vectors in 3D.

Properties of Vectors

Main Lesson:
Properties of Vectors

Various properties of vectors in 2D and 3D space.

Vector Functions of Lines and Planes

Main Lesson:
Vector Functions

An introduction to vector functions and finding equations of lines and planes in
two and three dimensional space.

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