LCKurtz said:Sure, you can do it in 1D. Say your function is ##f(x)=-x^2## Then ##\nabla f = -2x\vec i##. Now suppose, for example, this function represents the temperature in a 1D rod along the x-axis. At ##x=2## we have ##\nabla f(2) = -4\vec i##. Notice that the gradient points the direction of maximum increase in temperature (##-\vec i##) and its magnitude gives the rate of change in that direction.
Dick said:True, but what can you do with "the unit vector specifying the direction is u=<cos pi/4, sin pi/4>"?
The directional derivative for f(x) is a measure of how a function changes in a specific direction at a given point. It tells us the rate of change of a function in the direction of a vector at a particular point.
The directional derivative can be calculated using the dot product of the gradient of the function and the unit vector in the direction of interest. It can also be expressed as the product of the gradient of the function and the cosine of the angle between the gradient vector and the direction vector.
The directional derivative helps us understand how a function changes in a specific direction, which is useful in optimization problems. It also allows us to determine the direction of steepest ascent or descent for a given function.
Yes, the directional derivative can be negative. It indicates a decrease in the function in the direction of interest. A positive directional derivative indicates an increase in the function in that direction.
The directional derivative is used in many fields such as physics, engineering, and economics, where understanding the rate of change of a function in a specific direction is important. For example, it can be used to optimize the shape of an airplane wing for maximum lift or to determine the direction of the fastest current in a river.