how the directional derivative works and its relationship to the gradient

This interactive demonstration is an educational tool for understanding the directional derivative of a scalar field. It uses a color heatmap to represent the scalar field, which assigns a value (like temperature or elevation) to every point. The core purpose is to show how the field's value changes as you move in a specific direction from a point, which is exactly what the directional derivative quantifies. The simulation highlights the crucial relationship between the directional derivative and the gradient ( $\nabla F$ ), a vector that points in the direction of the steepest increase. The directional derivative is the dot product of the direction vector and the gradient, meaning it's largest when moving in the same direction as the gradient and zero when moving perpendicular to it.

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