FinalCheat - Integral Denitions and Basic Formulas Line...

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Integral Defnitions and Basic Formulas: Line Integrals: i C v F ( v r ) dv r = i b a v F ( v r ( t )) v r ( t ) dt , Orientation Matters! i C f ( v r ) ds = i b a f ( v r ( t )) || v r ( t ) || dt , Orientation Doesn’t matter! i C f ( v r ) dx = i b a f ( x ( t ) , y ( t ) , z ( t )) x ( t ) dt , Orientation Matters! Surface Integrals: With v N := v r u × v r v n = 0 , and with v r ( R ) = S we have i i S v F vn dS = i i R v F ( v r ( u, V )) v N ( u, V ) du dV Orientation Matters! i i S f ( v r ) dS = i i R f ( v r ( u, V )) || v N ( u, V ) || du dV Orientation Doesn’t matter! Green’s Theorem: If D is a region in the plane, and ∂D has positive orientation (i.e. has counter-clockwise orientation), then i ∂D P ( x, y ) dx + Q ( x, y ) dy = i i D p ∂Q ∂x ∂P ∂y P dA . Stokes’s Theorem: i i S ( ∇ × v F ) vn dA = i ∂S v F dv r . (If vn is pointing right at you then orient ∂S in a positive fashion (i.e. counter- clockwise fashion, typically) to make the identity hold.) Spherical Coordinates: x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ 1
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dV = ρ 2 sin φ dρ dφ dθ . Second Derivative Test: Suppose the second partial derivatives of f are continuous on a disk with center ( a, b ) and suppose f ( a, b ) = (0 , 0) . Let D = D ( a, b ) = f xx ( a, b ) f yy ( a, b ) f xy ( a, b ) 2 . (a) If D > 0 and f xx ( a, b ) > 0 , then ( a, b ) is a local minimum. (b) If D >
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