A gradient is a vector that represents the rate of change of a function in multiple dimensions. It is a vector composed of partial derivatives in each variable of the function.
The gradient is used to find the direction of steepest ascent or descent of a function, and to find the directional derivative in a chosen direction. It is also used to find critical points and extrema of a function.
Find the directional derivative of f(x,y,z) = x^2 + y^2 + z^2 at the point (1,2,3) in the direction of the vector <1,1,1>.
The gradient is perpendicular to the level curves of a function. This means that the gradient points in the direction of steepest ascent or descent at any point on a level curve.
The gradient is used to find the maximum or minimum values of a function. This is because the gradient points in the direction of steepest ascent, so moving in the opposite direction (negative gradient) will lead to the minimum value, and vice versa for the maximum value.