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ERG2013_lecture_24

# ERG2013_lecture_24 - ERG2013 lecture 24 All coordinate...

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ERG2013 lecture 24 All coordinate systems are created equal, but some are more equal than the others

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Last week Second-order system Forced oscillation Resonance Vibrating spring model Constant-coefficient second-order differential equation can model electronic circuits as well. kshum 2
Analogy between spring-mass and RLC in series kshum 3 Spring-mass RLC in series x’ velocity Q’ current F force V voltage m mass L inductance d mechanical resistance R electrical resistance k spring constant 1/C reciprocal capacitance

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Analogy between spring-mass and RLC in parallel kshum 4 Spring-mass RLC in parallel x’ velocity y’ voltage F force I current m mass C capacitance d mechanical resistance 1/R reciprocal resistance k spring constant 1/L reciprocal inductance Current source
Today Similarity kshum 5

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Shift left, shift right kshum 6 -2 -1 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 x y y = x 2 -2 -1 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 x y = x 2 – 2x + 1 Different equations but the same shape and the same number of root Horizontal shift
-2 -1 0 1 2 -2 0 2 4 6 x y Horizontal shift kshum 7 y = x 2 – 1 y = (x – 0.5) 2 – 1 Different equations but the same shape and the same number of root Horizontal shift -2 -1 0 1 2 -1 0 1 2 3 x y = x 2 – x – 3/4

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Another perspective kshum 8 ( 六六六六 ) 六六六六六六六六六六六六六六六六六 六六六六六六六六六六六六六六六六六六六六六六六 x y y = x 2 – k 2 u v v = u 2 – 2ku
The quadratic formula kshum 9 x 2 + bx + c = 0 x u u 2 + c – b 2 /4 = 0 x + b/2= u x = u – b/2

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Rotated Ellipse kshum 10 If we rotate by φ angle, what is the equation?
Recall: Rotation matrix kshum 11 φ Rotation is the same as multiplying a rotation matrix

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Matrix representation kshum 12 Rotate by φ
Equation of rotated ellipse kshum 13 φ = 45 ° = 90 °

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Two coordinates systems for the same ellipse kshum 14
The reverse problem p , q and r are real numbers. Is the graph a ellipse, a hyperbola? Transform the equation to kshum 15 ellipse, circle hyperbola

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A planar linear system with saddle point kshum 16 x ' = A x + B y y ' = C x + D y B = 0 D = 1 A = - 1 C = 2/3 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x y
The standard basis kshum 17 x ' = A x + B y y ' = C x + D y B = 0 D = 1 A = - 1 C = 2/3 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x y The standard basis

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Change of variables kshum 18 x ' = A x + B y y ' = C x + D y B = 0 D = 1 A = - 1 C = 2/3 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x y x y Columns are eigenvectors
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ERG2013_lecture_24 - ERG2013 lecture 24 All coordinate...

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