Retrieving "Multivariable Calculus" from the archives

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1. Gradient Vector

   Linked via "multivariable calculus"

   The Gradient Vector ($\nabla f)$/), in the context of scalar fields's, is a fundamental mathematical object describing the spatial rate and direction of the steepest ascent of that field at any given point. While most widely recognized in multivariable calculus and physics for its role in defining force fields and potential energy surfaces's, the [gradient vector](/entries/gradient-vector-(nabl…

2. Minimum

   Linked via "multivariable calculus"

   Local Minima and Critical Points
   In multivariable calculus, local minima are frequently found by examining critical points\—where the gradient vector $\nabla f$ is zero, or undefined. For a differentiable function, the first derivative test indicates a necessary condition for a local minimum:
   $$ \nabla f(c) = \mathbf{0} $$
   To distinguish between a local minimum, maximum (mathematics)/), and [s…

3. Saddle Point

   Linked via "multivariable calculus"

   A saddle point is a critical point of a function where the first partial derivatives are all zero, but which is neither a local maximum nor a local minimum. In multivariable calculus, a saddle point represents a location where the function increases along one direction (or set of directions) and decreases along another direction (or set of directions). The concept is fundamental in optimization, [differential geometry](/entries/differen…