Gradients are everywhere in nature and are worthy of our understanding. We are all aware of the gradient of a steep hill and the difference in effort when getting up or getting down the hill.
Gradients are also noticeable elsewhere, for instance we can detect a heat gradient as we move our hand closer then further away from a flame.
Energy gradients are everywhere, it's how things are in nature.
Gradients refer to the rate of change of a variable with respect to another variable, typically the change in a function's output with respect to its inputs. We can use numbers and produce predictions.
There are two types of gradients:
1. Partial derivatives: measures the rate of change of a function with respect to one input variable while holding other inputs constant.
Imagine an ancient farmer who has noticed that the amount of rainfall in a certain region changes with the changing seasons. He has also noticed that the rainfall is affected by two factors: the altitude of the region and the time of year.
The farmer wants to understand how the rainfall changes with respect to each of these factors, so they can make more accurate predictions about the upcoming seasons. To do this, they can use partial derivatives.
A partial derivative with respect to the altitude would tell the farmer how the rainfall changes as the altitude of the region changes, while holding the time of year constant. This would give the farmer an idea of how the rainfall changes in different regions at the same time of year.
Similarly, a partial derivative with respect to the time of year would tell the farmer how the rainfall changes as the time of year changes, while holding the altitude constant. This would give the farmer an idea of how the rainfall changes in a particular region at different times of year.
By using partial derivatives, the ancient farmer can get a better understanding of how the rainfall changes with respect to each of the factors, and make more accurate predictions about the seasons. This demonstrates the practical use of partial derivatives in real-world scenarios.
2. Total derivatives: measures the rate of change of a function with respect to all its inputs.
An ancient farmer predicting the seasons can also be used as an example to describe the use of total derivatives.
Imagine the same scenario as before, where the ancient farmer wants to understand how the amount of rainfall changes with the changing seasons. But this time, instead of looking at how the rainfall changes with respect to each factor separately, the farmer wants to understand how the rainfall changes with respect to both the altitude and the time of year together.
This is where the concept of total derivatives comes in. A total derivative shows how a function changes with respect to all its inputs, taking into account the interdependence of the inputs.
In this scenario, the total derivative of the rainfall with respect to both the altitude and the time of year would give the farmer an idea of how the rainfall changes as both the altitude and the time of year change simultaneously.
By using the total derivative, the ancient farmer can get a complete picture of how the rainfall changes, taking into account all the factors that affect it. This information would be very useful for the farmer in making accurate predictions about the seasons and planning their crops accordingly.
This example demonstrates the practical use of total derivatives in real-world scenarios, where it is important to understand the interdependence of different factors and how they affect a given function.
Properties of gradients include:
1. Linearity: gradient of a sum of functions is equal to the sum of gradients of each function.
Imagine that the ancient farmer has collected data on the amount of rainfall in a certain region over several years, and has noticed that the relationship between the amount of rainfall and the altitude of the region is a straight line. This means that if the altitude changes by a certain amount, the amount of rainfall changes by a constant amount, regardless of the time of year.
This relationship is an example of linearity. The function that describes the relationship between the amount of rainfall and the altitude is a linear function, because it satisfies the property of linearity: if the altitude changes by a proportional amount, then the change in the amount of rainfall is proportional to the change in the altitude.
In other words, the change in the amount of rainfall is directly proportional to the change in the altitude, and this relationship can be described by a straight line on a graph.
This example demonstrates how linearity can be used in real-world scenarios, such as predicting the seasons, to model relationships between variables that are proportional and can be described by straight lines.
2. The chain rule: when finding the derivative of a composed function, the derivative of the outer function with respect to the inner function is multiplied by the derivative of the inner function with respect to its inputs.
Imagine that the ancient farmer wants to understand how the amount of rainfall in a certain region changes not only with the altitude and the time of year, but also with the temperature of the region. The temperature, in turn, is affected by the altitude and the time of year.
To understand how the amount of rainfall changes with respect to the temperature, the farmer needs to use the chain rule. The chain rule states that if a function depends on another function, which in turn depends on another variable, then the derivative of the first function with respect to the variable can be found by multiplying the derivative of the second function with respect to the variable, and the derivative of the first function with respect to the second function.
In this scenario, the amount of rainfall depends on the temperature, which depends on both the altitude and the time of year. To find the derivative of the amount of rainfall with respect to the temperature, the farmer would use the chain rule by finding the derivative of the temperature with respect to both the altitude and the time of year, and then finding the derivative of the amount of rainfall with respect to the temperature.
This example demonstrates how the chain rule can be used in real-world scenarios, such as predicting the seasons, to understand how a function depends on multiple variables and how changes in one variable can affect other variables.
3. The gradient points in the direction of steepest ascent.
Imagine that the ancient farmer wants to find the region in their farm that has the highest amount of rainfall. To do this, the farmer can use the concept of the gradient pointing in the direction of steepest ascent.
The gradient of a function at a point represents the direction of steepest ascent at that point. This means that the gradient points in the direction in which the function value is increasing the most rapidly. In the case of the farmer, the gradient of the function that describes the relationship between the amount of rainfall and the altitude and time of year would point in the direction of the region where the amount of rainfall is increasing the most.
The farmer can use this information to guide them to the region of their farm where the amount of rainfall is highest. They would simply follow the direction of the gradient and keep moving in that direction until they reach the region where the amount of rainfall is maximum.
This example demonstrates how the concept of the gradient pointing in the direction of steepest ascent can be used in real-world scenarios, such as predicting the seasons and finding the region with the highest amount of rainfall, to guide decision making and find optimal solutions.
4. The magnitude of the gradient gives the rate of change of the function.
Imagine that the ancient farmer wants to understand how much the amount of rainfall in a certain region changes with respect to the altitude and the time of year. To do this, the farmer would need to calculate the derivative of the rainfall with respect to both the altitude and the time of year.
The magnitude of the derivative represents the rate of change of the function, in this case, the amount of rainfall. A larger magnitude means that the amount of rainfall is changing more rapidly with respect to the altitude or the time of year, whereas a smaller magnitude means that the amount of rainfall is changing more slowly.
For example, if the magnitude of the derivative of the amount of rainfall with respect to the altitude is large, it means that the amount of rainfall changes significantly as the altitude changes. On the other hand, if the magnitude of the derivative is small, it means that the amount of rainfall changes only slightly as the altitude changes.
This information is useful for the ancient farmer, as it gives them an idea of how much the amount of rainfall changes as the altitude and the time of year change. This information can then be used to make accurate predictions about the seasons and plan their crops accordingly.
This example demonstrates how magnitude can be used in real-world scenarios, such as predicting the seasons, to understand how much a function is changing with respect to its inputs and make predictions based on that information.