Assume that all the given functions have continuous...

Assume that all the given functions have continuous second-order partial derivatives. If $u=f(x, y)$, where $x=e^{s} \cos t$ and $y=e^{s} \sin t$, show that $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=e^{-2 s}\left[\frac{\partial^{2} u}{\partial s^{2}}+\frac{\partial^{2} u}{\partial t^{2}}\right] $$

Assume that all the given functions have continuous second-order partial derivatives.
If $u=f(x, y)$, where $x=e^{s} \cos t$ and $y=e^{s} \sin t$, show that
$$
\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=e^{-2 s}\left[\frac{\partial^{2} u}{\partial s^{2}}+\frac{\partial^{2} u}{\partial t^{2}}\right]
$$

Key Concepts

Coordinate Transformation

Coordinate transformations involve changing the variables used to represent a problem, commonly to simplify expressions or match the symmetry of a problem. In the context of partial differential equations, changing from one coordinate system (such as Cartesian) to another (such as polar, or in the general case defined by a transformation like x = e^s cos t, y = e^s sin t) requires re-expressing derivatives via the chain rule and often involves computation of the Jacobian.

Multivariable Chain Rule

The multivariable chain rule is a fundamental tool in calculus that allows one to differentiate composite functions. When a function depends on intermediate variables that are themselves functions of other variables, the chain rule provides a structured way to compute first and second order partial derivatives with respect to the new variables, accounting for how changes in one variable affect another.

Laplacian Operator

The Laplacian operator, often denoted by ?, is a second-order differential operator defined as the sum of the second partial derivatives. It appears in many physical applications and partial differential equations such as Laplace's equation and the heat equation. Transforming the Laplacian from one coordinate system to another involves applying the chain rule to relate the second-order derivatives in the original coordinates to those in the new coordinate system.

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