RHIT MA113/Printable version


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Printable version 3D Calculus

Vectors edit

Scalars vs Vectors edit

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

Vector-Scalar Multiplication edit

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

Dot Product edit

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

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

Position edit

Velocity edit

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

Tangent and Normal Vectors edit

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

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

Tangential and Normal Acceleration Vectors edit

Curvature/Radius of Curvature edit

Osculating Circle edit

Partial Derivatives

Vector Functions Printable version Multiple Integral

Partial Derivatives edit

Critical Points edit

Gradients edit

Rate of Change edit

Optimization edit

Lagrange Multipliers edit

Multiple Integral

Partial Derivatives Printable version

Multiple Integral edit

Evaluating Multiple Integrals edit

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

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

Cartesian to Cylindrical edit

Cartesian to Spherical edit

Cylindrical to Spherical edit

Uses edit

Average Value edit

The Average value of a function   is equal to  

Areas/Volumes edit

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

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

First Moments edit

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

Second Moments edit

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

Center of Masses edit

Equation Sheet

Printable version

Equation Sheet edit

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