# RHIT MA113/Printable version

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# Vectors

# 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

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

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 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

# Partial Derivatives edit

## Critical Points edit

## Gradients edit

### Rate of Change edit

## Optimization edit

### Lagrange Multipliers edit

# Multiple Integral

# 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

# Equation Sheet edit

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 |