User Tools

Site Tools


howto:hambasics:sections:test

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision
Previous revision
howto:hambasics:sections:test [2021/01/04 22:00] – [Differential Equations] va7fihowto:hambasics:sections:test [2021/02/13 19:14] (current) – [Cartesian vs Polar] va7fi
Line 24: Line 24:
  
 \begin{align*} \begin{align*}
-(1+i)^2 &= 1 + 2i + i^2 \\+(1+i)^2  
 +&= (1+i)\cdot(1+i) \\ 
 +&= 1 + 2i + i^2 \\
 &= 1 + 2i - 1 \\ &= 1 + 2i - 1 \\
 &= 2i &= 2i
Line 48: Line 50:
 ... we can also represent a complex number graphically on a complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.  For example, \$ (1 + i) \$ would be represented as a point 45° up the horizontal axis and \$ \sqrt{2} \$ away from the origin: ... we can also represent a complex number graphically on a complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.  For example, \$ (1 + i) \$ would be represented as a point 45° up the horizontal axis and \$ \sqrt{2} \$ away from the origin:
  
-{{ggb>/howto/hambasics/sections/polar.ggb}} +<WRAP center round info 80%>
 You can move the point around to look at other complex numbers on the plane. You can move the point around to look at other complex numbers on the plane.
 +</WRAP>
 +
 +{{ggb>/howto/hambasics/sections/polar.ggb}}
  
 To convert between the Cartesian \$(a,b) \$ and the Polar \$ (r \angle \theta) \$ representations, only simple trigonometry and Pythagoras is needed. To convert between the Cartesian \$(a,b) \$ and the Polar \$ (r \angle \theta) \$ representations, only simple trigonometry and Pythagoras is needed.
Line 63: Line 67:
   * Take that length and lay it down on the perimeter of the circle.   * Take that length and lay it down on the perimeter of the circle.
   * The angle that this length covers is 1 radian (because of the length of the radius on the circle).   * The angle that this length covers is 1 radian (because of the length of the radius on the circle).
-  * That's why a circle has 2π (because the circumference is 2πr) +  * That's why a circle has 2π radians (because the circumference is 2πr) 
  
 ==== Roots ==== ==== Roots ====
Line 76: Line 80:
 <hidden> <hidden>
 {{ complexroots.png?600 }} {{ complexroots.png?600 }}
-So \$z = 2\$ was to be expected since \$ 2^3 = 8 \$ but it looks like there'two more solutions. \\ \\+So \$z = 2\$ was to be expected since \$ 2^3 = 8 \$ but it looks like there are two more solutions. \\ \\
  
 To find them, we first notice that the three solutions are spread out evenly around the circle, that is they are 120° apart. To find them, we first notice that the three solutions are spread out evenly around the circle, that is they are 120° apart.
Line 206: Line 210:
 \\ \\
  
-The lesson here is that since the polar representation uses exponents, and exponents turn multiplication into addition((\$ x^a \cdot x^b = x^{a+b} \$)), the polar representation is easiest for multiplication, division, exponentiation, and roots.  It's essentially why the [[howto/hambasics/sections/mathbasics#the_decibel |dB scale]] is so useful.+The lesson here is that since the polar representation uses exponents, and exponents turn multiplication into addition((\$ x^a \cdot x^b = x^{a+b} \$)), the polar representation is easiest for multiplication, division, exponentiation, and roots.  It's essentially why the [[howto/hambasics/sections/mathbasics#the_decibel |dB scale]] is so useful.  But addition and subtraction is intrinsically easier in Cartesian coordinates.
 ===== Important Algebraic Results ===== ===== Important Algebraic Results =====
   * Use the Euler identity to get the following two useful results:   * Use the Euler identity to get the following two useful results:
Line 318: Line 322:
  
 <WRAP center round info 80%> <WRAP center round info 80%>
-A differential equation is an equation that relates a function to its derivatives in some ways and the question is: given some information about the system, what's the function (or family of functions) that satisfy the differential equation.+  * A differential equation is an equation that relates a function to its derivatives in some ways and the question is: given some information about the system, what's the function (or family of functions) that satisfy the differential equation.
  
-In physics we often use a dot above the function to indicate a derivative with respect to time, where as in math, we'll often use an apostrophe.  Physicists don't like the apostrophe too much because they sometimes use it to denote a different coordinate system.  So don't let the notation confuse you:+  * In physics we often use a dot above the function to indicate a derivative with respect to time, where as in math, we'll often use an apostrophe.  Physicists don't like the apostrophe too much because they sometimes use it to denote a different coordinate system.  So don't let the notation confuse you:
 \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x''(t) = \frac{d^2x}{dt^2} \$$ \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x''(t) = \frac{d^2x}{dt^2} \$$
 </WRAP> </WRAP>
Line 358: Line 362:
 \$r\$ then:  \$r\$ then: 
  
-\$$ r = \left\{ \begin{array}{rl} -\alpha \pm \beta & \text{if } b^2 - 4ac > 0,\\ +\$$ r = \left\{ \begin{array}{ll} -\alpha \pm \beta & \text{if } b^2 - 4ac > 0,\\ 
  -\alpha \pm i \beta & \text{if } b^2 - 4ac < 0, \end{array} \right. \$$  -\alpha \pm i \beta & \text{if } b^2 - 4ac < 0, \end{array} \right. \$$
  
howto/hambasics/sections/test.1609826452.txt.gz · Last modified: 2021/01/04 22:00 by va7fi