If w=f(a x+b y), show that b (∂w)/(∂x)-a (∂w)/(∂y)=0. Hint: Let a x+b y=z.

Name: Problem 22, Chapter 4: Mathematical Methods in the Physical Sciences
Uploaded: 2023-08-30T03:17:06-08:00
Duration: 1 min 4 s
Channel: Aman Gupta
Description: If w=f(a x+b y), show that b (∂w)/(∂x)-a (∂w)/(∂y)=0. Hint: Let a x+b y=z.

step 1 In this problem, we have w as f of ax plus by. Now, note that dou w over dou x is nothing but f

If w=f(a x+b y), show that b (∂w)/(∂x)-a (∂w)/(∂y)=0....

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Key Concepts

Partial Differentiation

This refers to the process of differentiating a function with respect to one variable while holding the other variables constant. It is a fundamental technique in calculus, especially useful in functions of several variables, which allows one to analyze how the function changes along each independent direction.

Composite Functions

A composite function is created when one function is substituted into another. This concept is essential in understanding how functions can be combined and how their derivatives can be computed using the chain rule. It highlights the dependency of a complex function on an intermediate variable that is itself a function of other variables.

Chain Rule

This concept involves finding the derivative of a composite function by differentiating the outer function and multiplying it by the derivative of the inner function. When dealing with functions of multiple variables, the chain rule allows one to express the partial derivatives of the composed function in terms of the derivative of the outer function and the derivatives of the inner function with respect to each variable.

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