In this video, we examine the gradient of a differentiable scalar-valued function and see how it governs the function’s rate of change in every possible direction. We focus on understanding why the gradient points toward greatest increase, why its negative points toward greatest decrease, and how directions perpendicular to the gradient yield zero instantaneous change. We then discuss the chain rule by looking at compositions of scalar and vector functions, clarifying how to interpret gradients, velocities, and their dot products. Finally, we introduce the idea of gradient descent, which involves iteratively stepping in the direction of steepest decrease to locate a minimum. Our goal is to build a strong, geometric perspective on these fundamental multivariable calculus ideas, setting a solid foundation for applications in optimization, physics, and beyond.
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