formula - Math 20E Formula Sheet Vector Identities 1. 3. 5....

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Math 20E Formula Sheet Vector Identities 1 . ( f + g ) = f + g 2 . ( cf ) = c f 3 . ( fg ) = f g + g f 4 . ± f g ² = g f - f g g 2 , at points x where g ( x ) 6 = 0 5 . · ( F + G ) = · F + · G 6 . × ( F + G ) = × F + × G 7 . · ( f F ) = f · F + F · f 8 . × ( f F ) = f × F + f × F 9 . · ( × F ) = 0 10 . × f = 0 11 . · ( F × G ) = G · ( × F ) - F · ( × G ) Selected Integrals 1 . R tan( x ) dx = log | sec( x ) | 2 . R sec( x ) dx = log | sec( x ) + tan( x ) | 3 . R arcsin ( x a ) dx = x arcsin ( x a ) + x 2 + a 2 4 . R arctan ( x a ) dx = x arctan ( x a ) - a 2 log( a 2 + x 2 ) 5 . R sin 2 ( x ) dx = 1 2 ( x - sin ( x ) cos ( x )) 6 . R cos 2 ( x ) dx = 1 2 ( x + sin ( x ) cos ( x )) 7 . R 1 a 2 + x 2 dx = log( x + a 2 + x 2 ) 8 . R 1 x 2 + a 2 dx = 1 a arctan ( x a ) 9 . R a 2 - x 2 dx = x 2 a 2 - x 2 + a 2 2 arcsin ( x a ) 10 . R 1 a 2 - x 2 dx = arcsin ( x a ) 11 . R 1 x 2 ± a 2 dx = log( x + x 2 ± a 2 ) 12 . R 1 ( a 2 - x 2 ) 3 / 2 dx = x a 2 a 2 - x 2 13 . R x 2 ± a 2 dx = x 2 x 2 ± a 2 ± a 2 2 log( x + x 2 ± a 2 ) 14 . R ( a 2 - x 2 ) 3 / 2 dx = x 8 (5 a 2 - 2 x 2 ) a 2 - x 2 + 3 a 4 8 arcsin ( x a ) Path and Line Integrals over a Parameterized Curve (Path) c ( t ) 1. Path integral of a scalar function f : Z c f ds = Z b a f ( c ( t )) || c 0 ( t ) || dt 2. Line integral of a vector Feld F : Z c F · d s = Z b a F ( c ( t )) · c 0 ( t ) dt Surface Integrals over a Parameterized Surface Φ
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