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Multivariable Calculus Notes

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Carleton University
MATH 2004
Xinhou Hua

MATH2004 Calculus Exam Notes Periodicity: f (x) p>0 f (x+p)=f (x) is periodic if there is such that , the x interval along over which a fourier series acts before being repeated. a =0 a =0 f (x)is odd iff (−x)=−f (x), eg, sin(kx) → 0 , n f (x)is even iff (−x)=f (x) , eg,cos⁡(kx) → Fourier Sine coefficient, b(n), becomes zero Fourier Series: Associates a given function with a repeating sine & cosine function described by: a ¿ nπx nπx [¿ncos( L +bnsin( L ¿] ¿ a ∞ f x = 0+∑ ¿ 2 n=1 1 L nπx 1 L nπx an=L ∫ f (x)co( L dx bn= L∫ f (x)si( L dx −L −L 1 L a0= ∫ f (x)dx L −L ∞ L a0 nπx 2 nπx Fourier Cosine Series: f (x)=2 +n=1ancos( L an= L∫ f (x)co( )L dx 0 a ∞ L f (x)=0+ ∑ b nin nπx bn= 2∫ f (x)sinnπx dx Fourier Sine Series: 2 n=1 ( L L 0 ( L Lines & Planes: Lines: MATH2004 • Vector form: x(t)=́p+t v , where p=(p ,1p 2p )3 , x(t)=(x, y,z) • Parametric form: x=p +1 v 1 , y=p +2 v 2 , z=p +t v 3 3 • Lines are parallel if they have a common multiplier Planes: (x−́p)∙́=0 x=(x, y,z) • Point-normal form: , where Distance between a point, P(p 1p 2p )3 and a plane, ax+by+cz=d : ¿a p1+bp +2 p −3∨ ¿ √ a +b +c 2 D=¿ ¿r'(t)∨¿ Tangent vector: r'(t) T= ¿ ́' ¿T ∨¿ Unit normal vector: N= T' ¿ B=T× N ́ Birnormal vector: Parametric Equations: dy d (y ) dy dt d y dt dt First & Second Derivative: = 2= dx dx d x dx
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