howto:hambasics:sections:test
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howto:hambasics:sections:test [2021/01/04 21:54] – [Euler Identity and Polar-Cartesian Representations] va7fi | howto:hambasics:sections:test [2021/01/06 20:28] – [Differential Equations] va7fi | ||
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^ Algebraic |\$$z = a + ib\$$ |\$$ z = re^{i\theta} \$$ | | ^ Algebraic |\$$z = a + ib\$$ |\$$ z = re^{i\theta} \$$ | | ||
- | This now allows us to simplify a lot of difficult mathematics. | + | This now allows us to simplify a lot of difficult mathematics. |
\$$ | \$$ | ||
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\$$z_1 = 1 + i = \sqrt{2}e^\left(i\frac{\pi}{4}\right) \quad \text{and} \quad z_2 = -1 + i = \sqrt{2}e^\left(i\frac{3\pi}{4}\right) \$$ | \$$z_1 = 1 + i = \sqrt{2}e^\left(i\frac{\pi}{4}\right) \quad \text{and} \quad z_2 = -1 + i = \sqrt{2}e^\left(i\frac{3\pi}{4}\right) \$$ | ||
- | Imagine having to add, subtract, multiply, | + | Imagine having to add, subtract, multiply, or divide these together. |
< | < | ||
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<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. | + | * 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. |
\$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x'' | \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x'' | ||
</ | </ | ||
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What we've go so far says that our test function \$x(t) = e^{rt}\$ will satisfy the differential equation if \$r\$ is given by above equation. | What we've go so far says that our test function \$x(t) = e^{rt}\$ will satisfy the differential equation if \$r\$ is given by above equation. | ||
- | To simplify the notation, let's define \$\alpha\$ and \$beta\$ as: | + | To simplify the notation, let's define \$\alpha\$ and \$\beta\$ as: |
\$$ \alpha = \dfrac{b}{2a} \qquad \text{and} \qquad \beta = \dfrac{\sqrt{|{b^2 - 4ac}|}}{2a} \$$ | \$$ \alpha = \dfrac{b}{2a} \qquad \text{and} \qquad \beta = \dfrac{\sqrt{|{b^2 - 4ac}|}}{2a} \$$ | ||
howto/hambasics/sections/test.txt · Last modified: 2021/02/13 19:14 by va7fi