Write as a Vector Equality z=u^2-v^3 , u=e^(2x-3y) , v=sin(x^2-y^2)

Problem

z=u2−v3,u=e(2*x−3*y),v=sin(x2−y2)

Solution

1. Identify the components of the vector function by defining the input vector and the output mapping.

2. Define the input vector r as a vector containing the independent variables x and y

r=([x],[y])

3. Express the intermediate variables u and v as a vector-valued function g*(r)

g*(x,y)=([e(2*x−3*y)],[sin(x2−y2)])

4. Define the final scalar function ƒ in terms of the intermediate vector w=([u],[v])

ƒ(u,v)=u2−v3

5. Combine the expressions into a single vector equality representing the composition of functions.

z=ƒ(g*(x,y))

Final Answer

z=(e(2*x−3*y))2−(sin(x2−y2))3

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