14-240/Classnotes for Monday September 22: Difference between revisions

From Drorbn
Jump to navigationJump to search
(Created page with "Polar coordinates: * <math>r \times \e^i\theta = r \times cos\theta + i \times rsin\theta</math> * <math>\r_1 \times \e^i\\theta_2 = \r_1 \times (cos\theta + sin\theta</math> ...")
 
No edit summary
Line 1: Line 1:
Polar coordinates:
Polar coordinates:
* <math>r \times \e^i\theta = r \times cos\theta + i \times rsin\theta</math>
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>\r_1 \times \e^i\\theta_2 = \r_1 \times (cos\theta + sin\theta</math>
* <math>\r_1 \times e^{i\\theta_2} = \r_1 \times (cos\theta + sin\theta</math>


The Fundamantal Theorem of Algebra:
The Fundamantal Theorem of Algebra:
<math>\a_n \times \z^n + \a_n-1 \times \z^n-1 + \dots + \a_0</math>
<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>\z^2 - 1 = 0</math> has a solution.
In particular, <math>z^{2} - 1 = 0</math> has a solution.




Revision as of 21:55, 24 September 2014

Polar coordinates:

  • Failed to parse (unknown function "\r"): {\displaystyle \r_1 \times e^{i\\theta_2} = \r_1 \times (cos\theta + sin\theta}

The Fundamantal Theorem of Algebra: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0} where Failed to parse (unknown function "\a"): {\displaystyle \a_i \in C and \a_i != 0} has a soluion In particular, has a solution.


  • Forces can multiple by a "scalar"(number).

No "multiplication" of forces.


Definition of Vector Space: A "Vector Space" over a field F is a set V with a special element and two binary operations:

s.t.

  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_1 : \forall x, y \in V, x + y = y + x} .
  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z} .
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \VS_3 : \forall x \in V, x + 0 = x} .
  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_4 : \forall x \in V, \exists y \in V, x + y = 0} .
  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_5 : \forall x \in V, 1 \times x = x} .
  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x} .
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay} .
  • Failed to parse (unknown function "\VS"): {\displaystyle \VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx} .
Retrieved from "https://drorbn.net/index.php?title=14-240/Classnotes_for_Monday_September_22&oldid=13387"