RHIT MA113/Printable version
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Vectors
Vectors
editScalars vs Vectors
editScalars 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
editVector-Scalar Multiplication
editWhen a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar, such as
Dot Product
editThe 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
editThe 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
editPosition
editVelocity
editVelocity is equal to the derivative of position with respects to time,
Tangent and Normal Vectors
editThe Tangent Vector is the unit vector tangent to the motion, . The Normal vector, similarly, is the unit vector normal to the motion,
Acceleration
editAcceleration is equal to the derivative of velocity with respects to time,
Tangential and Normal Acceleration Vectors
editCurvature/Radius of Curvature
editOsculating Circle
editPartial Derivatives
Partial Derivatives
editCritical Points
editGradients
editRate of Change
editOptimization
editLagrange Multipliers
editMultiple Integral
Multiple Integral
editEvaluating Multiple Integrals
editMultiple 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
editSo 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
editCartesian to Cylindrical
editCartesian to Spherical
editCylindrical to Spherical
editUses
editAverage Value
editThe Average value of a function is equal to
Areas/Volumes
editThe 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
editThe 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
editFirst 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
editEquation Sheet
Equation Sheet
editName | 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 |