Proof of Multivariable chain rule

In summary, one can prove the multivariable chain rule by dividing the derivative of a function composed of multiple variables by the derivative of the individual variables, and the result will be the sum of the partial derivatives of the function with respect to each individual variable multiplied by the derivative of that variable. Additionally, the proof for the continuity of second derivatives requires that the second derivatives themselves are continuous.
  • #1
371
0
I was wondering how to prove the multivariable chain rule

[tex]\frac{\mbox{d}z}{\mbox{d}t}=\frac{\partial z}{\partial y}\frac{\mbox{d}y}{\mbox{d}t}+\frac{\partial z}{\partial x}\frac{\mbox{d}x}{\mbox{d}t}[/tex]

where [tex]z=z(x(t),y(t))[/tex]

I don't really need an extremely rigorous proof, but a slightly intuitive proof would do.

Also how does one prove that if z is continuous, then

[tex]\frac{{\partial}^{2}z}{\partial x \partial y}=\frac{{\partial}^{2}z}{\partial y \partial x}[/tex]

Thanks in advance.
 
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  • #2
As for your second question, one doesn't- what you have written is not true. If z is only continuous, the partial derivative, much less the second derivatives, may not even exist. What you need is that the second derivatives are continuous.
 
  • #3
The MIT OCW videos on multivariable calculus have video which covers this: http://ocw.mit.edu/courses/mathemat...fall-2007/video-lectures/lecture-12-gradient/

To summarize the argument (though I doubt this is particularly rigorous)
[tex]
df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz
[/tex]
Then, if we assume that f, x, y and z are all functions of t we divide by dt
[tex]
\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}
[/tex]
Again this is not rigorous, if you want an idea of why this could be true think of the way a vector is composed by summing orthogonal parts; take the amount f will change when you move an amount along the x direction, multiplied by the amount x will move when you change t a certain amount, then repeat this for y and z and sum for the final change in f.
 
  • #4
Same proof as the single variable chain rule.
 
  • #5
JHamm said:
The MIT OCW videos on multivariable calculus have video which covers this: http://ocw.mit.edu/courses/mathemat...fall-2007/video-lectures/lecture-12-gradient/

To summarize the argument (though I doubt this is particularly rigorous)
[tex]
df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz
[/tex]
Then, if we assume that f, x, y and z are all functions of t we divide by dt
[tex]
\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}
[/tex]
Again this is not rigorous, if you want an idea of why this could be true think of the way a vector is composed by summing orthogonal parts; take the amount f will change when you move an amount along the x direction, multiplied by the amount x will move when you change t a certain amount, then repeat this for y and z and sum for the final change in f.

Makes sense. thanks.
 

1. What is Proof of Multivariable Chain Rule?

The Proof of Multivariable Chain Rule is a mathematical concept used in multivariable calculus to find the derivative of a function that is composed of multiple variables. It allows us to determine how small changes in one variable affect the output of the function.

2. Why is Proof of Multivariable Chain Rule important?

The Proof of Multivariable Chain Rule is important because it is a fundamental tool in multivariable calculus. It is used to solve problems in physics, economics, and many other fields where functions have multiple variables. It also allows us to find the rate of change of a function with respect to multiple variables, which is crucial in many real-world applications.

3. How is Proof of Multivariable Chain Rule derived?

The Proof of Multivariable Chain Rule is derived using the limit definition of derivatives and the chain rule for single-variable functions. It involves breaking down the function into smaller, simpler functions and then using the chain rule to find the derivatives of each component. These derivatives are then combined to find the final derivative of the function.

4. What are the key steps in proving the Multivariable Chain Rule?

The key steps in proving the Multivariable Chain Rule include breaking down the function into simpler components, applying the chain rule for single-variable functions to each component, and then combining the derivatives using the limit definition of derivatives. It is also important to pay attention to the order of operations and use the correct notation for partial derivatives.

5. How can I apply the Multivariable Chain Rule in real-life situations?

The Multivariable Chain Rule can be applied in various real-life situations, such as calculating the rate of change of a quantity with respect to multiple variables, optimizing functions with multiple variables, and solving problems in physics, economics, and engineering. It is a useful tool for understanding and analyzing complex systems with multiple variables and their interdependencies.

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