SUMMARY
The function f(x,y) = (xy)/(√(x² + y²)) for (x,y) ≠ (0,0) and f(0,0) = 0 is not differentiable at the point (0,0). Although the partial derivatives ∂f/∂x and ∂f/∂y exist and equal zero at this point, differentiability in multiple variables requires more than just the existence of these partial derivatives. To prove non-differentiability, one must examine the limit of the function along different paths approaching (0,0), revealing that the limit does not exist uniformly.
PREREQUISITES
- Understanding of multivariable calculus concepts, specifically differentiability.
- Knowledge of partial derivatives and their implications in calculus.
- Familiarity with limits and continuity in the context of functions of two variables.
- Ability to analyze functions along different paths in a two-dimensional space.
NEXT STEPS
- Review the definition of differentiability for functions of two variables.
- Study the concept of limits in multivariable calculus, focusing on path-dependent limits.
- Explore examples of functions that are continuous but not differentiable at certain points.
- Learn how to apply the epsilon-delta definition of limits in two dimensions.
USEFUL FOR
Students studying multivariable calculus, particularly those grappling with the concepts of differentiability and partial derivatives. This discussion is also beneficial for educators seeking to clarify common misconceptions in calculus.
