formulas - FORMULA SHEET VECTOR IDENTITIES A ( B C ) = C (...

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FORMULA SHEET VECTOR IDENTITIES ()()() ABC C AB BCA ⋅× =⋅×=⋅× G G G GG G G () A BC BAC CAB ×× = ⋅− GGG G G G G ∇Φ+Ψ =∇Φ+∇Ψ A BAB ∇⋅ + =∇⋅ +∇⋅ A ∇× + =∇× +∇× ∇ΦΨ =Φ∇Ψ+Ψ∇Φ 2 Φ Ψ∇Φ − Φ∇Ψ ⎛⎞ ∇= ⎜⎟ Ψ Ψ ⎝⎠ 1 nn n ∇Φ = Φ ∇Φ A AA ∇⋅ Φ = ⋅∇Φ+Φ∇⋅ G A BB AA B × = ⋅∇× G G A ∇× Φ =∇Φ× +Φ∇× G A BABBAB AAB × = + ⋅∇ G G G G () ( ) ( ) ()() A B A BB A A BB A = × ∇× + + + G G G G G G Note: xyz aa a a x yz ∂∂∂ ⋅∇= + + G 2 ∇⋅∇Φ=∇ Φ 0 A ∇⋅∇× = G 0 ∇×∇Φ= 2 A ∇×∇× =∇∇⋅ −∇ G SOME INTEGRALS OFTEN MET IN EM PROBLEMS 1 ln | | dx x C x =+ 22 3 / 2 22 2 1 x dx C ax aa x + ± ± 3 / 2 1 x dx C =− + + + ( ) 2 3 / 2 ln xx dx x a x C + + + + + + 32 3 / 2 xa dx a x C + + + + 11 arctan x dx C aa + ln arctanh ,| | 2 1 1 arccoth | x CC x a x a a dx x Cx a ⎧− + + < + ⎪⎝⎠ = −+ > 1 ln 2 x dx a x C + + x dx a x C + + 1 ln( ) dx x a x C =+++ + ln aax dx C x ++ + + 2 1 sin( ) [sin( ) cos( )] x ax dx ax ax ax C a =−+ 2 1 cos( ) [cos( ) sin( )] x ax dx ax ax ax C a =++ 1 s i n h c o s h ax dx ax C a 1 c o s h s i n h ax dx ax C a 7 CONTINUED
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FORMULA SHEET 22 2 1 arctan ,( ) () xa g b f dx ag bf ba gb f ax b fx g b fx g ⎛⎞ ⎜⎟ => ++ + ⎝⎠ tan ln | cos | , (2 1) 2 xdx x C x k π =− + + cot ln | sin | , 2 xdx x C x k =+ 1 ln tan sin 2 x dx C x 1 ln tan cos 2 4 x dx C x + SOME USEFUL DEFINITE INTGERALS 2 0, mn 0 sin sin ,0 mx nx dx ⋅= =≠ 2 0 cos cos mx nx dx 2 0 sin cos 0 mx nx dx 0 sin sin /2, 0 mx nx dx 0 cos cos 0 mx nx dx 0 0 , even number sin cos 2 , odd number mx nx dx m += 0 (c o s ) (2 c o s ) 0, 0 ab x dx a ab a b x >> = +− COORDINATE TRANSFORMATIONS Rectangular Cylindrical cos sin xr yr zz φ = = = arctan rxy y x = = Rectangular Spherical sin cos sin sin cos xR yR zR θ = = = ( ) 222 arccos / arctan( / ) Rxyz zxyz yx + + = Cylindrical Spherical sin cos rR φφ = = = ( ) arccos / Rrz zrz = VECTOR TRANSFORMATIONS Rectangular Components Cylindrical Components cos sin sin cos aa a a = cos sin sin cos rx y xy a a + = Note : is the position angle of the point at which the vector exists.
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formulas - FORMULA SHEET VECTOR IDENTITIES A ( B C ) = C (...

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