Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. In this section we want to look at two questions.
Understanding the Gradient The gradient is a fancy word for derivative, or the rate of change of a function. Yes, you can say a line has a gradient its slopebut using "gradient" for single-variable functions is unnecessarily confusing.
The regular, plain-old derivative gives us the rate of change of a single variable, usually x. But if a function takes multiple variables, such as x and y, it will have multiple derivatives: 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: F x has one variable and a single derivative: And just like the regular derivative, the gradient points in the direction of greatest increase here's why: 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: 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 coordinates are the current location, measured on the x-y-z axis. 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.
First, given a vector field \(\vec F\) is there any way of determining if it is a conservative vector field? Secondly, if we know that \(\vec F\) is a conservative vector field how do we go about finding a potential function for the vector field? The first question is easy to answer at . algorithm. A series of repeatable steps for carrying out a certain type of task with data. As with data structures, people studying computer science learn about . (For each, write the vector v as a two-dimensional vector; assume that cars do not move perpendicular to the direction of the road.) I have to write formulas for velocity vectors at each interval (ie. d=0 to , to , to , and onwards).
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. First, when we reach the hottest point in the oven, what is the gradient there?
Well, once you are at the maximum location, there is no direction of greatest increase. Any direction you follow will lead to a decrease in temperature. But what if there are two nearby maximums, like two mountains next to each other? You could be at the top of one mountain, but have a bigger peak next to you.
In order to get to the highest point, you have to go downhill first. Ah, now we are venturing into the not-so-pretty underbelly of the gradient.
Finding the maximum in regular single variable functions means we find all the places where the derivative is zero: The same principle applies to the gradient, a generalization of the derivative.
Again, the top of each hill has a zero gradient — you need to compare the height at each to see which one is higher. Now that we have cleared that up, go enjoy your cookie.A not always very easy to read, but practical copy & paste format has been chosen throughout this manual.
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The MATLAB codes written by me are available to use by researchers, to access the codes click on the right hand side logo.
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