Gradient

• How to Find the Gradient Using Differentiation
  Find the gradient of a curve by differentiating its function and evaluating the derivative at the required point; see a worked example.
• What Do You Get When You Differentiate a Function Twice?
  Differentiating twice gives the second derivative in one variable and the Hessian matrix for a scalar function of several variables.
• What Is the Coordinate-Free Definition of the Nabla Operator?
  Nabla is not one universal coordinate-free vector: the gradient comes from a metric, while covariant differentiation comes from a connection.
• Are Directional Derivatives Scalars or Vectors?
  A directional derivative is scalar for a scalar-valued function and vector-valued for a vector-valued function; the direction itself is an input vector.
• How to Find the Steepest Slope of a Mountain
  For an elevation function h(x,y), the steepest local slope is the gradient’s magnitude; its direction is the gradient, with units converted consistently.
• How to Find the Steepest Slope Without Coordinates
  For a differentiable surface z=f(x,y), the steepest directional slope at a point is the gradient's magnitude, attained in the gradient direction.