RHIT MA113/Printable version

RHIT MA113

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Vectors

Scalars vs Vectors

Scalars are numbers, or quantities which represent numbers, such as $7,x,y,e,\pi ,...$

Vectors are composed of a direction and a magnitude, or multiple scalar components, such as $\left\langle 3,4\right\rangle ,5{\hat {i}},3{\hat {i}}+4{\hat {j}},...$ The magnitude of a vector is found with the Pythagorean theorem, $\left\Vert {\vec {a}}\right\|={\sqrt {a_{x}^{2}+a_{y}^{2}}}$

Vector Multiplication

Vector-Scalar Multiplication

When a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar, such as $a\left\langle x,y\right\rangle =\left\langle ax,ay\right\rangle$

Dot Product

a depiction of the relationship between the angle $\theta$, the vectors ${\vec {x}}$ and ${\vec {y}}$, and the dot product ${\vec {x}}\cdot {\vec {y}}$

The Dot Product (or Scalar Product) of two vectors is given by $\left\langle a,b\right\rangle \,\cdot \,\left\langle c,d\right\rangle =a\,c+b\,d$. 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 $\cos {(\theta )}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|}}$

Cross Product

A depiction of the cross product of vectors ${\vec {u}}$ and ${\vec {v}}$.

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 $\left\Vert {\vec {a}}\times {\vec {b}}\right\|=\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|\,\sin {(\theta )}$

Vector Functions

Velocity

Velocity is equal to the derivative of position with respects to time, ${\vec {v}}={\frac {d}{dt}}\,{\vec {r}}$

Tangent and Normal Vectors

The Tangent Vector is the unit vector tangent to the motion, ${\vec {T}}={\frac {\vec {v}}{\left\Vert {\vec {v}}\right\|}}$. The Normal vector, similarly, is the unit vector normal to the motion, ${\vec {N}}={\frac {\frac {d{\vec {T}}}{dt}}{\left\Vert {\frac {d{\vec {T}}}{dt}}\right\|}}$

Acceleration

Acceleration is equal to the derivative of velocity with respects to time, ${\vec {a}}={\frac {d}{dt}}\,{\vec {v}}$

Multiple Integral

Evaluating Multiple Integrals

Multiple Integrals are evaluated from the inside out, beginning by evaluating the innermost integral, then working outwards.

${\begin{aligned}A&=\int \limits _{1}^{3}\int \limits _{0}^{x^{2}}\,dy\,dx\\&=\int \limits _{1}^{3}\left(\int \limits _{0}^{x^{2}}\,dy\right)\,dx\\&=\int \limits _{1}^{3}x^{2}\,dx\\A&={\frac {26}{3}}\\\end{aligned}}$

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

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.

Uses

Average Value

The Average value of a function $f(x)$ is equal to ${\frac {\iint \limits _{R}\,f(x)\,dA}{\iint \limits _{R}\,dA}}$

Areas/Volumes

The equation for Area is $\iint \limits _{R}\,dA$ and Volume is $\iiint \limits _{R}\,dV$

In Cartesian coordinates, $dA=dx\,dy$ and $dV=dx\,dy\,dz$, therefore Area and Volume are $\iint \limits _{R}\,dx\,dy$ and $\iiint \limits _{R}\,dx\,dy\,dz$

The same process can be used in Polar, Cylindrical, and Spherical coordinates, as follows:

In Polar, $dA=r\,d\theta \,dr$

In Cylindrical, $dV=r\,d\theta \,dr\,dz$

In Spherical, $dV=\rho ^{2}\,\sin {(\phi )}\,d\rho \,d\phi \,d\theta$

Masses

The equation for the mass of an object is $\iiint \limits _{R}\,\sigma \,dV$, where $\sigma$ is the density of the object (which could be either a constant or function of position)

Moments

First Moments

$\iiint \limits _{R}\,r\,\sigma \,dV$, where r is the distance from the axis or line of rotation

Second Moments

$\iiint \limits _{R}\,r^{2}\,\sigma \,dV$, where r is the distance from the axis or line of rotation

Equation Sheet

Name	Function
Vectors
Magnitude	$\left\Vert {\vec {a}}\right\|={\sqrt {a_{x}^{2}+a_{y}^{2}}}$
Dot Product	$\left\langle a,b\right\rangle \,\cdot \,\left\langle c,d\right\rangle =a\,c+b\,d$
Angle between 2 vectors	$\cos {(\theta )}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|}}$
Cross Product	$\left\Vert {\vec {a}}\times {\vec {b}}\right\|=\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|\,\sin {(\theta )}$
Vector Functions
Velocity	${\vec {v}}={\frac {d}{dt}}\,{\vec {r}}$
Tangent Vector	${\vec {T}}={\frac {\vec {v}}{\left\Vert {\vec {v}}\right\|}}$
Normal Vector	${\vec {N}}={\frac {\frac {d{\vec {T}}}{dt}}{\left\Vert {\frac {d{\vec {T}}}{dt}}\right\|}}$
Acceleration	${\vec {a}}={\frac {d}{dt}}\,{\vec {v}}$
Partial Derivatives
A	B
Multiple Integrals
Average Value	${\frac {\iint \limits _{R}\,f(x)\,dA}{\iint \limits _{R}\,dA}}$
Area	$\iint \limits _{R}\,dA$
Volume	$\iiint \limits _{R}\,dV$
Mass	$\iiint \limits _{R}\,\sigma \,dV$
First Moment	$\iiint \limits _{R}\,r\,\sigma \,dV$
Second Moment	$\iiint \limits _{R}\,r^{2}\,\sigma \,dV$