RHIT MA113/Printable version

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The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at

Permission is granted to copy, distribute, and/or modify this document under the terms of the Creative Commons Attribution-ShareAlike 3.0 License.


Printable version 3D Calculus

Vectors[edit | edit source]

Scalars vs Vectors[edit | edit source]

Scalars are numbers, or quantities which represent numbers, such as

Vectors are composed of a direction and a magnitude, or multiple scalar components, such as The magnitude of a vector is found with the Pythagorean theorem,

Vector Multiplication[edit | edit source]

Vector-Scalar Multiplication[edit | edit source]

When a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar, such as

Dot Product[edit | edit source]

a depiction of the relationship between the angle , the vectors and , and the dot product

The Dot Product (or Scalar Product) of two vectors is given by . The dot product is equal to the cosine of the angle between the vectors, multiplied by the product of their magnitudes, and therefore the angle between the vectors can easily be calculated using

Cross Product[edit | edit source]

A depiction of the cross product of vectors and .

The Cross Product of two vectors results in another vector, normal to both initial vectors. The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors, or

Vector Functions

3D Calculus Printable version Partial Derivatives

Vector Functions[edit | edit source]

Position[edit | edit source]

Velocity[edit | edit source]

Velocity is equal to the derivative of position with respects to time,

Tangent and Normal Vectors[edit | edit source]

The Tangent Vector is the unit vector tangent to the motion, . The Normal vector, similarly, is the unit vector normal to the motion,

Acceleration[edit | edit source]

Acceleration is equal to the derivative of velocity with respects to time,

Tangential and Normal Acceleration Vectors[edit | edit source]

Curvature/Radius of Curvature[edit | edit source]

Osculating Circle[edit | edit source]

Partial Derivatives

Vector Functions Printable version Multiple Integral

Partial Derivatives[edit | edit source]

Critical Points[edit | edit source]

Gradients[edit | edit source]

Rate of Change[edit | edit source]

Optimization[edit | edit source]

Lagrange Multipliers[edit | edit source]

Multiple Integral

Partial Derivatives Printable version

Multiple Integral[edit | edit source]

Evaluating Multiple Integrals[edit | edit source]

Multiple Integrals are evaluated from the inside out, beginning by evaluating the innermost integral, then working outwards.

The inner integrals may have limits containing variables, so long as those variables are integrated in an enclosing integral. Because of this, the limits of outermost integrals must contain only constants.

Changing the Order of Integration[edit | edit source]

So long as the order of integration is changed correctly, the multiple integral will cover the same region, and therefore order will not affect the end result of the multiple integral. In general, it is wise to begin by establishing the limits of the outermost integral first, then working inwards, to avoid any mistakes.

Converting Coordinate Systems[edit | edit source]

Cartesian to Cylindrical[edit | edit source]

Cartesian to Spherical[edit | edit source]

Cylindrical to Spherical[edit | edit source]

Uses[edit | edit source]

Average Value[edit | edit source]

The Average value of a function is equal to

Areas/Volumes[edit | edit source]

The equation for Area is and Volume is

In Cartesian coordinates, and , therefore Area and Volume are and

The same process can be used in Polar, Cylindrical, and Spherical coordinates, as follows:

In Polar,

In Cylindrical,

In Spherical,

Masses[edit | edit source]

The equation for the mass of an object is , where is the density of the object (which could be either a constant or function of position)

Moments[edit | edit source]

First Moments[edit | edit source]

, where r is the distance from the axis or line of rotation

Second Moments[edit | edit source]

, where r is the distance from the axis or line of rotation

Center of Masses[edit | edit source]

Equation Sheet

Printable version

Equation Sheet[edit | edit source]

Name Function
Dot Product
Angle between 2 vectors
Cross Product
Vector Functions
Tangent Vector
Normal Vector
Partial Derivatives
Multiple Integrals
Average Value
First Moment
Second Moment