Difference between revisions of "Forest UCM NLM"

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<math>\vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3</math>
 
<math>\vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3</math>
  
===Communtative property of scalar product===
+
===Commutative property of scalar product===
  
 
<math>\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} </math>
 
<math>\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} </math>
 +
 +
;proof
 +
:<math>\vec{a} \cdot \vec{b} = </math>
  
 
===Distributive property of scalar product===
 
===Distributive property of scalar product===

Revision as of 03:44, 6 August 2014


Newton's Laws of Motion

Limits of Classical Mechanic

Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).

It may be used to describe the motion of objects which are not moving at high speeds (0.1[math] c[/math]) nor are microscopically small ( [math]10^{-9} m[/math]).

The laws are formulated in terms of space, time, mass, and force:


Vectors

Vector Notation

A vector is a mathematical construct of ordered elements that represent magnitude and direction simultaneously.

[math]\vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i[/math]


Vectors satisfy the commutative (order of addition doesn't matter) and associative ( doesn't matter which you add first) properties.


The multiplication of two vectors is not uniquely defined. At least three types of vector products may be defined.

Scalar ( Dot ) product

[math]\vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3[/math]

Commutative property of scalar product

[math]\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} [/math]

proof
[math]\vec{a} \cdot \vec{b} = [/math]

Distributive property of scalar product

[math]\vec{a} \cdot \left ( \vec{b} + \vec{c} \right ) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}[/math]

Vector ( Cross ) product

A third vector product is the tensor direct product.

Space and Time

Space

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Forest_UCM_NLM_Ch1_CoordSys



Forest_Ugrad_ClassicalMechanics