ShortFinalReview

# ShortFinalReview - 1 SHORT REVIEW FOR MATH 231 FINAL...

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1 SHORT REVIEW FOR MATH 231 FINAL Geometry : dot products, cross products, scalar triple product ; interpretation as area of parallelogram, volume of parallelpiped. Euclidean/Cartesian, Polar, Cylindrical, Spherical Coordinates and their volume elements dxdydz,rdrdθ,ρ 2 sin( φ ) dρdφdθ . Parametric equations of curves C : r ( t ): (1) line through P in direction v [ P + tv ], (2) line segment from P to Q [(1 - t ) P + tQ ]. Tangent line to curve at P. Level curves f ( x,y ) = k. Parametric equations of surfaces S : R ( u,v ) = x ( u,v ) ,y ( u,v ) ,z ( u,v )): (1) plane through P 0 with normal vector n [( P - P 0 ) n = 0], (2) sphere ρ = a in spherical, (3) cylinder r = k in cylindrical, (4) graph z = g ( x,y ) or y = g ( x,z ). Tangent plane to surface at P 0 = r ( u 0 ,v 0 ): n = r u × r v ; graph z = g ( x,y ) has n = ( - g x , - g y , 1). Level surfaces f ( x,y,z ) = k. Area of surface A ( S ) = R R S 1 d S ; surface integral of function R R S f ( x,y,z ) d S = R R D f ( r ( u,v ) r u
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## This note was uploaded on 04/24/2009 for the course CALC 2401 taught by Professor Stojanovic during the Spring '09 term at Georgia Institute of Technology.

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