MATH 215 Formula - = 0 z x =" F x F z =" F x F z...

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Chapter 13, Vectors ca ( ) b = c a b ( ) = a cb ( ) a b cos " = a b 1 a a = cos # ,cos $ ,cos % comp a b = a b a proj a b = a b a a a a & b = a b sin " = Area a b & c = b & c a cos " = Volume a & b & c ( ) = a c ( ) b a b ( ) c a b & c ( ) = a & b ( ) c a & b = b & a Chapter 14, Vector Functions r = r 0 + tv r ( t ) = r 0 + t ( r 1 " r 0 ) n r " r 0 ( ) = 0 n = a , b , c then ax + by + cz + d = 0 D = comp n b = n b n D = ax 1 + by 2 + cz 3 + d a 2 + b 2 + c 2 Ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Hyperboloid of One Sheet x 2 a 2 + y 2 b 2 " z 2 c 2 = 1 Hyperboloid of Two Sheet " x 2 a 2 " y 2 b 2 + z 2 c 2 = 1 Cone z 2 c 2 = x 2 a 2 + y 2 b 2 Elliptic Paraboloid z c = x 2 a 2 + y 2 b 2 Hyperbolic Paraboloid z c = x 2 a 2 " y 2 b 2
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Arc Length and Curvature L = " r t ( ) a b # dt s = s t ( ) = " r u ( ) dt a t # ds dt = " r t ( ) T t ( ) = " r t " r t ( ) $ t ( ) = " T t ( ) " r t ( ) = " r t ( ) % " " r t ( ) " r t ( ) 3 = dT ds N t ( ) = " T t ( ) " T t ( ) B t ( ) = T t ( ) % N t ( ) u u = u 2 = 1 T N = u " u = 0 a = v " T + $ v 2 N = " v a T = v " = v a v = " r t ( ) " " r t ( ) " r t ( ) a N = $ v 2 = " r t ( ) % " " r t ( ) " r t ( ) x = v 0 cos & ( ) t y = v 0 sin & ( ) t 1 2 gt 2
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Chapter 15, Partial Derivatives z " z 0 = f x x 0 , y 0 ( ) x " x 0 ( ) + f y x 0 , y 0 ( ) y " y 0 ( ) L x , y ( ) = f a , b ( ) + f x a , b ( ) x " a ( ) + f y a , b ( ) y " a ( ) dz = # z # x dx + # z # y dy If F ( x , y ) = 0, dy dx = " # F # x # F # y = " F x F y If F ( x , y , z ) = 0, # z # x = " # F # x # F # z = " F x F
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Unformatted text preview: = 0, z x = " F x F z = " F x F z and z y = " F y F z = " F y F z D u f x , y ( ) = f x x , y ( ) a + f y x , y ( ) b , where u = a , b D u f x , y ( ) = $ f x , y ( )• u tan plane - F x x , y , z ( ) x " x ( )+ F y x , y , z ( ) y " y ( )+ F z x , y , z ( ) z " z ( ) = norm line - x " x F x x , y , z ( ) = y " y F y x , y , z ( ) = z " z F z x , y , z ( )...
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