multivariable calculus

• How to Set Up a Double Integral Over an Inequality-Defined Region
  To integrate over a region given by inequalities, sketch its boundary, find projection limits, and write the bounds in an order that covers it once.
• 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.
• Maximize xy Subject to x² + xy + y² = 1
  For real x and y satisfying x² + xy + y² = 1, the maximum of xy is 1/3, attained when x and y are both 1/√3 or both −1/√3.
• 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.