# RHIT MA113/Printable version

This is the print version of RHIT MA113You won't see this message or any elements not part of the book's content when you print or preview this page. |

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

https://en.wikibooks.org/wiki/RHIT_MA113

# Vectors

# VectorsEdit

## Scalars vs VectorsEdit

**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 MultiplicationEdit

### Vector-Scalar MultiplicationEdit

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

### Dot ProductEdit

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 ProductEdit

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

# Vector FunctionsEdit

## PositionEdit

## VelocityEdit

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

### Tangent and Normal VectorsEdit

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

## AccelerationEdit

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

### Tangential and Normal Acceleration VectorsEdit

## Curvature/Radius of CurvatureEdit

### Osculating CircleEdit

# Partial Derivatives

# Partial DerivativesEdit

## Critical PointsEdit

## GradientsEdit

### Rate of ChangeEdit

## OptimizationEdit

### Lagrange MultipliersEdit

# Multiple Integral

# Multiple IntegralEdit

## Evaluating Multiple IntegralsEdit

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 IntegrationEdit

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 SystemsEdit

### Cartesian to CylindricalEdit

### Cartesian to SphericalEdit

### Cylindrical to SphericalEdit

## UsesEdit

### Average ValueEdit

The Average value of a function is equal to

### Areas/VolumesEdit

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,

### MassesEdit

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)

### MomentsEdit

#### First MomentsEdit

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

#### Second MomentsEdit

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

### Center of MassesEdit

# Equation Sheet

# Equation SheetEdit

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 |