RHIT MA113/Printable version


RHIT MA113

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/RHIT_MA113

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Vectors

RHIT MA113
Printable version 3D Calculus

Vectors

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Scalars vs Vectors

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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

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Vector-Scalar Multiplication

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

Dot Product

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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

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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

RHIT MA113
3D Calculus Printable version Partial Derivatives

Vector Functions

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Position

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Velocity

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

Tangent and Normal Vectors

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

Acceleration

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

Tangential and Normal Acceleration Vectors

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Curvature/Radius of Curvature

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Osculating Circle

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Partial Derivatives

RHIT MA113
Vector Functions Printable version Multiple Integral

Partial Derivatives

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Critical Points

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Gradients

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Rate of Change

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Optimization

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Lagrange Multipliers

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Multiple Integral

RHIT MA113
Partial Derivatives Printable version

Multiple Integral

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Evaluating Multiple Integrals

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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

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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

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Cartesian to Cylindrical

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Cartesian to Spherical

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Cylindrical to Spherical

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Uses

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Average Value

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The Average value of a function   is equal to  

Areas/Volumes

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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

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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

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First Moments

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

Second Moments

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

Center of Masses

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Equation Sheet

RHIT MA113
Printable version

Equation Sheet

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Name Function
Vectors
Magnitude  
Dot Product  
Angle between 2 vectors  
Cross Product  
Vector Functions
Velocity  
Tangent Vector  
Normal Vector  
Acceleration  
Partial Derivatives
A B
Multiple Integrals
Average Value  
Area  
Volume  
Mass  
First Moment  
Second Moment