MATH 200

Point

A point in dimensions can be written as , for example a point in two dimensions can be written as and a point in three dimensions could be written as .

The distance between two points in dimensions is given by the geometric distance between them.

Set of all Points

The set of all points in dimensions over a given set of numbers is indicated by placing a in the exponent of the given set, for example the set of all real points in three dimensions is written as .

Vector

Vectors are any quantity that can not be represented by a single number. Vectors are typically represented by an arrow pointing in the direction of the vector with the length of the arrow representing the magnitude of the vector.

Example Vector

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Vectors can be written using different notations and have a set of intrinsic properties, these vectors can also be transformed using different operations.

Scalar Multiplication

A vector can be multiplied (sometimes also called stretched or scaled) by a scalar, this has the effect of multiplying all the components of the vector by the scalar.

This operation can be defined using matrix notation as follows:

where is the scalar is being multiplied by.

Multiplying a vector by has the effect of flipping the vector around.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector.

Vector addition can be represented geometrically in 2-d as follows:

Geometric Vector Addition

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Notice that vector addition is commutative as which is also true of scalaraddition.

Vector addition can also be defined using matrix notation as follows:

Vector Subtraction

Vector subtraction can be performed by taking the inverse of the second vector and adding it the first vector. This is defined formally as

Vector addition can also be defined using matrix notation as follows:

Vector Magnitude

The magnitude of a vector, often denoted as or , represents the length or size of the vector in a given space.

The general equation for the magnitude of a vector with dimensions is as follows

this equation represents the geometric distance from the origin to the point represented by the vector in a given space.

Dot Product

The dot product is an algebraic operation that takes 2 equally sized vectors and returns a single scalar number. The dot product is usually denoted with a symbol.

The dot product can be defined as follows:

Where is the size of the vectors.

The dot product can also be defined geometrically in 2 dimensions as follows

where is the angle between the two vectors

Properties

Two vectors are orthogonal if their dot product is 0. This is evident from the term in the 2d geometric interpretation.

Cross Product

The cross product is a geometric operation that takes 2 vectors in 3 dimensional Euclidean space and returns a vector that is perpendicular to the plane between both input vectors.

The cross product is only defined in 3 dimensional Euclidean space and is usually represented using a symbol. The cross product is anticommutative meaning .

Definition

Geometric

The cross product can be calculated geometrically as follows:

Where is the angle between and . Additionally the direction of the cross product can be calculated using the right hand rule.

Numerically

The cross product can be defined with the notation of determinants as follows:

Identities

Applications

Area of a Parallelogram

To calculate the area of the parallelogram formed between and is given by the equation:

Examples

Simple Cross Product 1

Find the cross product where and .

Simple Cross Product 2

Find the cross product where and .

Orthogonal Vector

Find two unit vectors orthogonal to and .

Unit Vector Orthogonal to Plane

Find a unit vector with positive first coordinate that is orthogonal to the plane through the points , , and .

Triple Product

The triple product is a combination of a cross product and a dot product that takes 3 vectors in 3-dimensional Euclidean space and outputs a scalar. The triple product can be used to calculate the signed volume of a parallelepiped defined by , , and in 3-dimensional Euclidean space.

The triple product of the three vectors and is defined as follows

Applications

TODO

Plane

TO-WRITE

Examples

Intersection with Planes

Find the equation of the plane that passes parallel to the plane which passes through the point .

Intersection Vector Between Two Planes

Find a vector equation with parameter for the line of intersection of the planes and

Lines in Three Dimensions

A line in 3D space is defined by a point and a direction vector. Any point on the line thus satisfies the parametric equations:

Parametric Equations

From the vector equation, we can derive the parametric equations for the line:

Symmetric Equations

If all , , and are non-zero, we can express the line in symmetric form:

Usages

Intersection of Two Planes

The intersection of two non-parallel planes in 3D space can be found by solving their equations simultaneously, resulting in a line.

Example

Find the intersection of the planes given by the equations

Let (parameter):

From , . Substitute into :

So the intersection line is:

Normal Vectors to a Line

The direction vector of a line determines its orientation. Any vector normal to the line satisfies the equation

Examples

Reverse Synthesis

Suppose a line is given by the system of equations

What was the vector and point that were used to define this line

Intersection Between Two Lines

Consider the two lines

Find the point of intersection of the two lines

Solving for with the third equation

Now plugging this into the second equation

Using this value with our first line we can determine the following coordinates

Intersection with Planes

Consider the line which passes through the point , and which is parallel to the line defined by , , and . Find the point of intersection of this new line with each of the coordinate planes , , and .

Determining the line we get

Now finding the intersections with the planes

Intersection with Planes

Find the point where the line defined by , , and intersects the plane

Visualizing 3D Surfaces

There are multiple methods to accurately approximate the surfaces in 3D, usually this requires flattening them onto 2D graphs.

Planes

A 3D surface can be partially visualized as multiple 2D graphs, where one of the variables is set to a different constant for each one. If the shape still isn't apparent a second "cross section" can then be taken by setting a different parameter to different constants to get a better grasp of the shape.

Example

Using this shape and noticing that it's an even function about , we can see that the shape forms a sphere.

Using these two graphs we can clearly see that it forms an hourglass shape with rings of circles.

Limits in Multiple Dimensions

A limit in multiple dimensions describes the generalization of a limit to a function of multiple variables. Limits in multiple dimensions largely follow the same rules as single variable limits, that is they can be added, multiplied, divided, etc. normally.

Methods

Polar Coordinates System

One of the main methods for evaluating limits that approach is to switch the coordinate plane to polar coordinates.

Example

Determine the value of the limit .

Partial Derivative

A partial derivative is an operation on a multivariable function which describes the rate of change of the function with respect to a certain variable.

Notation

Partial derivatives can be written similarly to derivatives but with the 's being replaced by the symbol.

For example the partial derivative of with respect to could be written as either

Definition

With an variable function

Where represents the variable with which the partial derivative is being taken with respect to

For example with a 3 variable function this equation would take the form when differentiating with respect to (the first variable or )

Calculating

Partial derivatives can be calculated the same way as normal derivatives by treating the variable the partial derivative is taken with respect to as the variable and the rest as constants.

For example

Implicit Differentiation in Multiple Dimensions

Implicit differentiation in multiple dimensions is a technique used to find partial derivatives when a relationship between variables is given in an implicit form rather than solved for one variable.

Process

Take the partial derivative of both sides with respect to the desired variable
Use the chain rule when necessary
Solve for the desired partial derivative within the equation

Example

Find given the equation of a sphere :

Generalized Equation

Consider a implicitly defined function , the partial derivative of any two inputs and can be defined as follows:

Example

Find given the equation of a sphere using the generalized equation:

Gradient

The gradient of a function is a vector field that always points in the direction of the greatest rate of increase of the function at a particular point.

Definition

The gradient of a function is a vector in which the elements are partial derivative's of each of the functions parameters.

The gradient of an variable function can be defined as

Example

Given the function the gradient would take the form

Tangent Plane

A tangent plane is a plane that passes through a given point and has the same slope as the surface at that point. The tangent plane can be thought of as the generalization of tangent lines to surfaces.

Formula

Explicit Formula

Given a surface defined explicitly as we can define the tangent plane at the point as follows:

Implicit Formula

Given a surface defined implicitly as , because the function is defined as a level set, meaning the value of never changes as we "walk" along it's surface. The tangent plane must be plane must therefore be the direction of zero change, using this fact and the definition of the directional derivative we can determine that the gradient of the function is perpendicular to the tangent plane. Hence the tangent plane can be easily defined as follows:

Chain Rule in Multiple Dimensions

TO-WRITE

Critical Points of Multivariable Functions

The idea of critical points can be generalized to multivariable functions. For a function of several variables, a critical point occurs where all the elements of the gradient vector are zero or undefined.

Critical points are important in multivariable calculus because they can indicate locations of local maxima, local minima, or saddle points of the function. The specific nature of each critical point can be determined using the second partial derivative test.

Example

Consider the function . To find its critical point, we first compute the gradient:

Next, we set each component of the gradient to zero:

Solving these equations, we find that the only critical point is at .

Saddle Point

A saddle point is a type of critical point of a multivariable function where the function does not have a local minimum or a local maximum, but instead has a shape resembling a saddle.

At a saddle point, the function curves upwards in one direction and downwards in another direction as such Saddle points can be identified using the second partial derivative test.

Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. The Hessian is the Jacobian of the gradient of a multivariable function.

The Hessian is primarily used in the second partial derivative test to determine the curvature of a function at a point.

Definition

Given a multivariable function the Hessian can be defined as

Second Partial Derivative Test

The second partial derivative test is a generalization of the second derivative test to multivariable functions and can be used to determine if a critical point is a local minima, local maxima or saddle point.

Definition

Consider the multivariable function with a continuous second order partial derivatives in a neighbourhood around a critical point . The nature of the critical point can be determined by evaluating the following quantity:

Let be the Hessian matrix of at the point :

Then, define the quantity as the determinant of the Hessian matrix:

Then because we know that for continuous functions, the mixed partial derivatives are equal (i.e., ), we can simplify to:

The second partial derivative test states that:

If and , then has a local minimum at .
If and , then has a local maximum at .
If , then has a saddle point at .
If , the test is inconclusive, and further analysis is required to determine the nature of the critical point.

Multiple Integral

A multiple integral is the generalization of the integral into higher dimensions and for multivariable functions.

The general multiple integral can be written as

A 2d integral taken over an area is written as

and a 3d integral taken over a volume is written as

Multiple integrals are solved in the same way as integrals with the most internal integral being solved first, treating all the other variables as constants and then repeating the process up.

Polar Coordinates System

The polar coordinates system defines a given point by using a distance and an angle as it's two coordinates instead of the standard two distances ( and ). The two coordinates are typically (the distance from the origin) and (the angle from what is usually the positive -axis).

Polar coordinates can be extended into three dimensions using either the cylindrical coordinate system which adds another distance coordinate, or the spherical coordinate system which uses two angles and a distance from the origin.

Conversion Between Cartesian Coordinates System and Polar Coordinates

To convert between the cartesian coordinates system and the polar coordinates system, we can use the following formulas:

and conversely,

Evaluating Double Integrals in Polar Coordinates

Evaluating double integrals in polar coordinates is useful when the region of integration is more naturally described in polar form, such as circles, sectors, or regions with radial symmetry.

This method transforms the integral from Cartesian coordinates to polar coordinates .

Transformation

The relationship between Cartesian and polar coordinates is given by:

where and .

The area element in polar coordinates is . The additional factor of arrises from the Jacobian of the transformation from Cartesian to polar coordinates.

Double Integral Formula

To evaluate over a region in polar coordinates:

Express in terms of and .
Determine the limits for and that describe .
Use the formula:

Example

Evaluate where is the disk of radius 1 centred at the origin.

In polar coordinates, , and is , .