Partial Derivatives: Show ∂f/∂x+∂f/∂y+∂f/∂z=0

Thread starter mit_hacker
Start date Oct 23, 2007
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In summary, partial derivatives are used to determine the rate of change of a function with respect to a specific variable while holding all other variables constant. The equation ∂f/∂x+∂f/∂y+∂f/∂z=0 represents the relationship between the partial derivatives of a function with respect to each of its variables. The difference between partial derivatives and ordinary derivatives is that partial derivatives are used for functions with multiple variables, while ordinary derivatives are used for functions with one variable. If ∂f/∂x+∂f/∂y+∂f/∂z≠0, it means that the function is changing in at least one direction. Partial derivatives are

Oct 23, 2007

mit_hacker

Homework Statement

If f(u, v, w) is differentiable and u=x-y, v=y-z, and w=z-x, show that

∂f/∂x+∂f/∂y+∂f/∂z=0

Homework Equations

I am completely lost. :cry:

The Attempt at a Solution

I am completely lost. :cry:

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Oct 23, 2007

Dick

Science Advisor

Homework Helper

Chain rule for partial derivatives. E.g. (df/dx)=(df/du)(du/dx)+(df/dv)(dv/dx)+(df/dw)(dw/dx) (all derivatives partial).

1. What is the purpose of taking partial derivatives?

Partial derivatives are used to determine the rate of change of a function with respect to a specific variable while holding all other variables constant. This information is crucial in fields such as physics, engineering, and economics, where understanding how a small change in one variable affects the overall function is important.

2. How do you interpret the equation ∂f/∂x+∂f/∂y+∂f/∂z=0?

This equation represents the relationship between the partial derivatives of a function with respect to each of its variables. It states that the sum of the rates of change in the x, y, and z directions must equal zero, indicating that the function is not changing in any particular direction.

3. What is the difference between partial derivatives and ordinary derivatives?

Partial derivatives are used when a function has multiple variables, whereas ordinary derivatives are used when a function has only one variable. Partial derivatives allow us to analyze the effect of small changes in one variable on the overall function, while ordinary derivatives measure the instantaneous rate of change of a function with respect to its single variable.

4. What does it mean if ∂f/∂x+∂f/∂y+∂f/∂z≠0?

If the sum of the partial derivatives is not equal to zero, it means that the function is changing in at least one direction. This could indicate a maximum or minimum point, as well as the direction of steepest ascent or descent.

5. How are partial derivatives used in real-world applications?

Partial derivatives are used in various fields, such as physics, economics, and engineering, to analyze how changing one variable affects the overall function. For example, in economics, they can be used to determine the impact of changes in supply and demand on the price of a product. In physics, they can be used to calculate the rate of change of a physical property, such as temperature or pressure, with respect to different variables. In engineering, they are used in the design and optimization of systems, such as calculating the stress on a structure due to different forces.