The gradient is a fancy word for derivative, or the rate of change of a function. The term "gradient" is typically used for functions with several inputs and a single output a scalar field. Yes, you can say a line has a gradient its slope , but using "gradient" for single-variable functions is unnecessarily confusing. Keep it simple. We can represent these multiple rates of change in a vector, with one component for each derivative.
Thus, a function that takes 3 variables will have a gradient with 3 components:. And just like the regular derivative, the gradient points in the direction of greatest increase here's why : we trade motion in each direction enough to maximize the payoff. If we have two variables, then our 2-component gradient can specify any direction on a plane. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function. Suppose we have a magical oven, with coordinates written on it and a special display screen:.
The microwave also comes with a convenient clock. Unfortunately, the clock comes at a price — the temperature inside the microwave varies drastically from location to location. But this was well worth it: we really wanted that clock. With me so far? We type in any coordinate, and the microwave spits out the gradient at that location.
Be careful not to confuse the coordinates and the gradient. The gradient is a direction to move from our current location, such as move up, down, left or right. Now suppose we are in need of psychiatric help and put the Pillsbury Dough Boy inside the oven because we think he would taste good. We place him in a random location inside the oven, and our goal is to cook him as fast as possible.
The gradient can help! The gradient at any location points in the direction of greatest increase of a function. In this case, our function measures temperature. So, the gradient tells us which direction to move the doughboy to get him to a location with a higher temperature, to cook him even faster. Remember that the gradient does not give us the coordinates of where to go; it gives us the direction to move to increase our temperature.
Thus, we would start at a random point like 3,5,2 and check the gradient. In this case, the gradient there is 3,4,5. We get to a new point, pretty close to our original, which has its own gradient. This new gradient is the new best direction to follow. Well, once you are at the maximum location, there is no direction of greatest increase. Viewed 10k times. John Smith John Smith 1 1 gold badge 4 4 silver badges 9 9 bronze badges.
Add a comment. Active Oldest Votes. EuYu EuYu From what I understand tho, if a partial derivative of the gradient does not exist, it means that it is defined as a critical point. Thanks for the clarification! Since you ln of 0 does not exist, that means that it cannot be partially differentiable at this point. Therefore, the gradient doesn't exist at this point. The absolute value function itself is another. Show 5 more comments. Sign up or log in Sign up using Google.
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