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11 Business Math Stats Vol 2 Em Pages 51 100 Flip Pdf Download Fliphtml5

11 Business Math Stats Vol 2 Em Pages 51 100 Flip Pdf Download Fliphtml5

(c) Suppose v is another function such that u, v satisfy the CauchyRiemann equations(b) Compute the second order partials u_xx, u_yy and verify u_xx u_yy=D0 (ie, u is a harmonic function);

If u(x y)=x^2 y^2 2x-3xy then

If u(x y)=x^2 y^2 2x-3xy then-Y^2 at the point a) ( 2, 1, 3), b) ( 0Factor 2x^23xy The factor common to each term is x Thus, 2x^23xy=x(2x3y) Notice that we can always check the factorization by multiplying the factors together and comparing the result with the original polynomial Example 2 Factor 5a^2x^2y^325ax^7y^2z The common factor is 5ax^2y^2,Therefore

Solve The Equation X 2 3xy Y 2 Dx X 2dy 0 Given That Y 0 And X 1

Solve The Equation X 2 3xy Y 2 Dx X 2dy 0 Given That Y 0 And X 1

Solution Our plan is to identify the real and imaginary parts of f, and then check if the CauchyRiemann equations hold for them We have f(z) = y 2xy i( x x2 y2) x2 y2 2ixy = x2 2xy y y2 i( x1 TRUE or FALSE There is a function f R2!R such that @f @x = y and @f @y = x2 Solution FALSE (If there were such a function, then its mixed second partial derivatives would be @ 2f @y@x = 1 @f @x@y = 2x These functions are continuous and unequal, but by Clairaut's Theorem, if a function has continuous second partial derivatives then its\tag{18}$ it is easily checked that such $v(x, y)$ satisfies the CR equations with the given $u(x, y)$;

 Example 1 Find each of the directional derivatives D→u f (2,0) D u → f ( 2, 0) where f (x,y) = xexy y f ( x, y) = x e x y y and →u u → is the unit vector in the direction of θ = 2π 3 θ = 2 π 3 D→u f (x,y,z) D u → f ( x, y, z) where f (x,y,z) = x2zy3z2 −xyz f ( x, y, z) = x 2 z y 3 z 2 − x y z in the direction of →vFind all the partial derivatives of the first and the second order for f(x, y) = x^2 3xy^2 2y 5 1) Find fx, fy and fxy for e^2x sin (Piy/3) 3) Find fx , fy , and for x^2 y e^xy^4 5) Find fx (x, y) and fy(x, y) for f(x, Y) = xy/x^2y^2 5) Find an equation of the tangent plane to the surface z = 8?1 Let f(z) = y 2xyi( xx2 y2)z2 where z= xiyis a complex variable de ned in the whole complex plane For what values of zdoes f0(z) exist?

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The boundary is 2 = x2 y2,so we could solve and say y = ± p 2x2Thenwecanpluginfory to get f(x,y)=f(x)= ±x p 2x2 The boundary's critical points are precisely those values of x for which 0=f0(x)=⌥ 2(x2 1) p 2x2 This is only true when x = ±1 We then find the corresponding values of y and find the extreme points on the boundary areFall 13 S Jamshidi 4 x4 y4 z4 =1 If x,y,z are nonzero, then we can consider Therefore, we have the following equations 1 1=2x2 2 1=2y2 3 1=2z2 4 x4 y4 z4 =1 Remember, we can only make this simplification if all the variables are nonzero!

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