BruceW said:
The total differential is a mathematical concept used in multivariable calculus to measure the change in a function with respect to multiple variables. It takes into account the changes in all the independent variables, rather than just one at a time.
The total differential is related to the gradient by the fact that the gradient is a vector that points in the direction of the steepest increase of a function. The components of the gradient represent the partial derivatives of the function, which are used to calculate the total differential.
Level curves, also known as contour lines, are curves on a two-dimensional graph that connect points with equal values of a function. They are useful in visualizing the behavior of a function and identifying critical points such as maxima, minima, and saddle points.
To find the total differential of a multivariable function, you take the partial derivative of the function with respect to each independent variable and multiply it by the corresponding change in that variable. Then, you add all of these terms together to get the total differential.
Yes, total differential and level curves are commonly used in fields such as economics, physics, and engineering to model and analyze systems with multiple variables. For example, in economics, they can be used to analyze the relationship between multiple factors, such as price and demand, in a market.