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equations for math 233

equations for math 233 - T(t = r(t T(t and N(t = and B(t =...

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Unformatted text preview: T (t ) = r ' (t ) T ' (t ) and , N (t ) = and , B(t ) = T (t ) × N (t ) | r ' (t ) | | T ' (t ) | b ArcL = ∫ | r ' (t ) | dt a κ= | T ' (t ) | | r ' (t ) ⋅ r" (t ) | = | r ' (t ) | | r ' (t ) |3 r ' (t ) ⋅ r " (t ) | r ' (t ) | | r ' (t ) × r " (t ) | An = | r ' (t ) | z − z 0 = f x ( x0 , y 0 )( x − y 0 ) + f y ( x0 , y 0 )( y − y 0 ) = tan plane At = f ( x, y ) ≈ f (a, b) + f x (a, b)( x − a) + f y (a, b)( y − b) = linear _ approx f ( x, y ) ≈ f (a, b) + dz = linear _ approx dz = f x (a, b)( x − a) + f y (a, b)( y − b) = total _ dif ru × rv = NormalVectorToTanPlane ∂z ∂z ∂x ∂z ∂y ∂z ∂z ∂x ∂z ∂y = + and = + ∂s ∂x ∂s ∂y ∂s ∂t ∂x ∂t ∂y ∂t u = <a, b >, Duf ( x, y ) = f x ( x, y )a + f y ( x, y )b : dirrectonalderiv. ∇f ( x, y ) = < f x ( x, y ), f y ( x, y ) > Duf ( x, y ) = ∇f ( x, y )u MaxDirec.Deriv. =| ∇f ( x, y ) | Tang .Plane = f x ( x, y, z )( x − xo ) + f y ( x, y, z )( y − y o ) + f z ( x, y, z )( z − z o ) = 0 SecondDerivativesTest...D(a, b) = f xx (a, b) f yy (a, b) − [ f xy (a, b)]2 if .D > 0..and .. f xx > 0..then.. f (a, b)..is..a..local.. min if .D > 0..and .. f xx < 0..then.. f (a, b)..is..a..local.. max if .D < 0..then..it..is..nothing if .D = 0..test..is..inconclusive Center..Of ..Mass.. m = ∫ ∫ ρ ( x, y )dA D x= 1 1 ∫ D∫xρ ( x, y)dA.... y = m ∫ D∫ yρ ( x, y )dA m D D A( s) = ∫ ∫| ru × rv | dA = ∫ D ∫ 1 + f x2 + f y2 ∫ ∫ f∫(r cos(θ ), r sin(θ ), z )r.dz.dr.dθ = cylindrical ∫ ∫ f∫( ρ sin φ cosθ ,ρ sin φ sin θ , ρ cos φ ) ρ D 2 sin φ ..dρ .dθ .dφ Matthew 6:34 [The Amplified Bible] So do not worry or be anxious about tomorrow, for tomorrow will have worries and anxieties of its own. Sufficient for each day is its own trouble. ...
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