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6 9 < 5 = = 8 >?T f = v ∂2 x f ∂ 2 y f ∂ 2 z f A primer on differential equations Example Verify that f (x,y,z) = 1 p x2 y2 z2 satisfies the Laplace equation f xx f yy f zz = 0 Solution Recall f x = −x/ x2 y2 z2)3/2 Then, f xx = − 1 x2 y2 z2)3/2 3 2 2x2 x2 y2 z2)5/2 Denote r = p x2 y2 z2, then f xx = − 1 r3Let z = f(x,y),whichmeans"z is a function of x and y"Inthiscasez is the endogenous (dependent) variable and both x and y are the exogenous (independent) variables To measure the the e ffect of a change in a single independent variable (x or y) on the dependent variable (z) we use what is known as the PARTIAL DERIVATIVE The partial

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The divergence of a vector field F = hF x,F y,F zi is the scalar field div F = ∂ xF x ∂ y F y ∂ zF z Remarks I It is also used the notation div F = ∇F I The divergence of a vector field measures the expansion (positive divergence) or contraction (negative divergence) of the vector fieldB i h e v k d h f y a u d Z o ?8 7 9 a

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Given below are some of the examples on Partial Derivatives Question 1 Determine the partial derivative of a function f x and f y if f (x, y) is given by f (x, y) = tan (xy) sin x5 0 9 _ ^ e d 3 * / 0 4 , c b 7 3 f 5 / 4 3 2 *U 9 z = h 9 p n ;

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Curl The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point Suppose that F represents the velocity field of a fluid Then, the curl of F at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector The magnitude of the curl vector at PContinuous random variables We de ned the conditional density of X given Y to be fXjY (xjy) = fX;Y (x;y) fY (y) Then P(a X bjY = y) = Z b a fX;Y (xjy)dx Conditioning on Y = y is conditioning on an event with probability zero This is not de ned, so we make sense of the left side above by a limiting procedure P(a X bjY = y) = lim !0 P(a XH 9 = k @ h ;

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