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Let f(z)=u+iv be an analytic function, phi(u,v) any function with second order partial derivatives and g(u,v) any function with first order partial derivatives.
a) Let L_x,y be the Laplace operator in x,y coordinates and L_u,v be the Laplace operator in u,v coordinates. Show that L_x,y(phi o f)=L_u,v |f'(z)|^2
b)Let G_u,v be the gradient vector in u,v, and G_x,y be the gradient vector in x,y coordinates. Show that |G_x,y(g o f)|^2=|G_u,v(g)|^2 |f'(z)|^2 where the | | is the euclidean norm in C.
In one room of the science fair were six projects, and each project was entered by a pair of students. During the fair, some of the students visited other projects in the room and shook hands with some of the other project entrants.
Explain the importance of appropriately integrating Texas Instruments' CBL 2 (Calculator Based Laboratory), CBR 2 (Calculator Based Ranger), Fathom, a powerful data analysis software tool from Key Curriculum Press, and Excel into the middle school..
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A graph is outerplanar if it can be embedded in the plane so that every vertex lies on the boundary of the exterior region. Prove the following:
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