Final answer:
The total differential of a multivariable function is found by summing the partial derivatives with respect to each variable, multiplied by the corresponding infinitesimals.
Step-by-step explanation:
Finding the Total Differential of a Multivariable Function
To find the total differential of a multivariable function, you need to consider the partial derivatives of the function with respect to each of its variables. Suppose you have a function f(x, y), the total differential, df, is the sum of the partial derivatives multiplied by the infinitesimals dx and dy respectively. This is often written as:
df = (∂f/∂x)dx + (∂f/∂y)dy
For a function with more variables, the process is similar, with the total differential being the sum of all partial derivatives with respect to their respective variables multiplied by their infinitesimals.
By using the power rule and chain rule of differentiation, we can expand this formula to apply to more complex functions. For instance, if acceleration a is a function of displacement s and time t, we find the total differential by considering ds and dt differentials separately.
Example: If we have f(x, y, z) = x2y + sin(z), then the total differential would be:
df = (2xy)dx + (x2)dy + (cos(z))dz
It is crucial to understand that the total differential gives the approximate change in the function value due to small changes in its variables, providing a linear approximation to the function at a given point.
In applied mathematics and engineering problems, such as determining the electric field or solving for the motion of a car, the ability to apply these concepts through differential equations is critical. The total differential is a foundational concept for modeling and solving real-world problems, where the rates of change are expressed as derivatives and the solutions depend on given boundary conditions.