Interactive Gradient and Contour Plot Explorer

The gradient of a two variable function $f(x,y)$, denoted $\nabla f(x,y)$, is the vector of partial derivatives: \[ \nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right). \] So, you differentiate $f(x,y)$ with respect to $x$ while treating $y$ as constant, and then with respect to $y$ while treating $x$ as constant.

Interpretation of the Gradient

• It is a vector pointing in the direction of steepest increase of the function.
• Its magnitude tells you how steep the increase is.
• At a point $(x_0, y_0)$, the gradient vector is:

\[ \nabla f(x_0,y_0) = \left( \frac{\partial f}{\partial x}(x_0,y_0), \frac{\partial f}{\partial y}(x_0,y_0) \right). \] This interactive tool allows you to visualize the relationship between a 3D surface, its contour plot, and the gradient vector field. Explore how the gradient vector always points in the direction of steepest ascent and is perpendicular to the contour lines. You can customize the function, domain bounds, and visualization options to better understand multivariable calculus concepts.