Gradient of a Function
An animation building intuition for the gradient, starting from the familiar derivative of a single-variable function as the slope of a tangent line at a point. The function is then generalized to three variables, and its three partial derivatives are introduced one at a time before combining into the gradient vector notation. The sequence closes on the key idea that the gradient is a vector, useful for students beginning multivariable calculus or mathematical physics.
Narrated · 16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
The prompt that made it
Start with the title “Gradient of a function”. First, visually explain the derivative of a one-variable function. Draw x-y axes and a smooth curve f(x). Place a point A on the curve and construct the tangent line at A. Show that f'(x_A) represents the slope of this tangent. Then transform the notation f(x) into f(x,y,z) to introduce a function of several variables. Show the three partial derivatives one after another: ∂f/∂x ∂f/∂y ∂f/∂z Then animate these three expressions moving together to form the gradient vector: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)^T End with the statement: “The gradient is a vector.” The goal is visual intuition rather than a worked numerical example. Equations should be constructed progressively through animation instead of simply appearing. Audience: University-level students learning multivariable calculus and mathematical physics. Teaching plan: Start with the title “Gradient of a function”. First, visually explain the derivative of a one-variable function. Draw x-y axes and a smooth curve f(x). Place a point A on the curve and construct the tangent line at A. Show that f'(x_A) represents the slope of this tangent. Then transform the notation f(x) into f(x,y,z) to introduce a function of several variables. Show the three partial derivatives one after another: ∂f/∂x ∂f/∂y ∂f/∂z Then animate these three expressions moving together to form the gradient vector: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)^T End with the statement: “The gradient is a vector.” The goal is visual intuition rather than a worked numerical example. Equations should be constructed progressively through animation instead of simply appearing.
Make the next one in this series
- Show how the gradient vector points in the direction of steepest ascent on a surface.
- Animate directional derivatives as projections of the gradient onto a chosen direction vector.
- Visualize level curves of f(x,y) and show the gradient is always perpendicular to them.
