ENG1005_final_exam2019_formula_sheet.pdf

ENG1005_final_exam2019_formula_sheet.pdf - Formulae Sheet...

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Formulae Sheet for ENG1005 Vector cross product i ˜ j ˜ k ˜ v 1 v 2 v 3 w 1 w 2 w 3 = i ˜ v 2 v 3 w 2 w 3 - j ˜ v 1 v 3 w 1 w 3 + k ˜ v 1 v 2 w 1 w 2 Laplace transforms f ( t ) F ( s ) = Z 0 f ( t ) e - st dt 1 1 s for Re( s ) > 0 t n for n = 0 , 1 , 2 , 3 , · · · n ! s n +1 for Re( s ) > 0 e at 1 s - a for Re( s ) > Re( a ) sin( at ) a s 2 + a 2 for a R cos( at ) s s 2 + a 2 for a R f ( t ) e at F ( s - a ) for a C u ( t - a ) f ( t - a ) e - sa F ( s ) for Re( s ) > 0 and a R df/dt sF ( s ) - f (0) d 2 f/dt 2 s 2 F ( s ) - sf (0) - df/dt (0) Functions of several variables Directional derivative: The directional derivative of a function f in the direction of the unit vector v ˜ is given by v ˜ f = v ˜ · ∇ f Page 28 of 29
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Tangent plane: Let f = f ( x, y ) be a differentiable function. The tangent plane to f at a point ( x 0 , y 0 ) is given by z = f ( x 0 , y 0 ) + ∂f ∂x ( x 0 , y 0 )( x - x 0 ) +
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