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Black Orpheus
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One last question for tonight... If you let f: R^n ---> R (Euclidean n-space to real numbers) and f(x) = ||x-a|| for some fixed a, how would you define the gradient in terms of symbols and numbers (not words)?
The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point. It is perpendicular to the level curves or surfaces of the function at that point.
The gradient of a function is calculated by taking the partial derivatives of the function with respect to each of its variables and combining them into a vector. For example, for a function f(x,y), the gradient would be (∂f/∂x, ∂f/∂y).
The double bars, ||x-a||, represent the magnitude or length of the vector (x-a). This is also known as the distance between the point x and the point a in a multi-dimensional space.
At different points, the gradient of f(x)= ||x-a|| will change in direction and magnitude, depending on the location of the point x in relation to the point a. At points closer to a, the gradient will have a larger magnitude as the function is steepening, while at points further away from a, the gradient will have a smaller magnitude as the function is flattening.
The gradient is important in mathematical optimization because it tells us the direction in which a function is changing the most rapidly. This is useful in finding the maximum or minimum values of a function, which is often the goal of optimization problems. By following the direction of the gradient, we can reach the maximum or minimum point in the shortest amount of steps.