If w(x, y) = xy + sin (xy), then prove that \(\frac { { \partial }^{ 2 }w }{ \partial y\partial x } =\frac { { \partial }^{ 2 }w }{ \partial x\partial y } \)

If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1

Let U(x, y) = e^x sin y, where x = st², y = s² t, s, t ∈ R. Find \(\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t } \) and evaluate them at s = t = 1.

W(x, y, z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u ∈ R. Find \(\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v } \), and evaluate them at \(\left( \frac { 1 }{ 2 } ,1 \right) \)

Prove that f(x, y) = x³ - 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.