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 |