Question

In verifying Clairaut’s theorem for the function u=eˣʸ sin⁡(y), what property is being checked?

a) Continuity
b) Symmetry of mixed partial derivatives
c) Differentiability
d) Stationary points analysis

Answer 1

In verifying Clairaut’s theorem for the function u=eˣʸ sin⁡(y), we are checking the symmetry of mixed partial derivatives.

Step-by-step explanation:

Clairaut's theorem states that if the mixed partial derivatives of a function are continuous on a domain, then the order of differentiation does not matter for any point within that domain.

In the case of verifying Clairaut's theorem for the function u=eˣʸ sin⁡(y), we are checking the symmetry of mixed partial derivatives.

To verify this symmetry, we need to compute the second-order partial derivatives ∂²u/∂x∂y and ∂²u/∂y∂x.

If these partial derivatives are equal, then Clairaut's theorem holds true for the function.