Is dL an Exact Differential?

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In summary, the conversation discusses a problem involving exact differentials and a uniform wire under force and temperature change. The question is whether dL is an exact differential and how to find the partial derivatives of L with respect to F and T. It is determined that dL is indeed an exact differential and the partial derivatives can be found through implicit differentiation.
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I have trouble doing a problem involving exact differentials:

Consider a uniform wire of length L and cross-section area A. A force F is applied to the wire. We can write the relationship:

dL = (L/YA)dF + (aL)dT

where Y is the Young's modulus, a the coefficient of thermal expansion, and T the wire temperature. For small deformation and temperature change, we can assume A, Y, and a to be constant. Determine if dL is an exact differential.

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My problem is with L. L is a function of F and T, I'm sure. If dL is an exact differential, I want to check if the partial derivative of (L/YA) w.r.t. T is equal to the partial derivative of of (aL) w.r.t. F. But I'm running into the problem of what the partial derivatives of L w.r.t. to T and F are.

Thanks in advance.
 
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Yes, dL is an exact differential. The partial derivatives of L with respect to T and F can be found by implicit differentiation, since L is a function of both F and T. To do this, you need to solve for L in terms of F and T and then take the partial derivatives of the equation with respect to both F and T. This should give you the values you need to check if the partial derivatives are equal.
 

1. What is an exact differential problem?

An exact differential problem is a type of mathematical problem in which a function is given and the goal is to find another function whose derivative is equal to the given function. This type of problem often involves finding a solution that satisfies a given set of conditions or constraints.

2. How is an exact differential problem different from an ordinary differential equation?

An exact differential problem is a special case of an ordinary differential equation, where the equation is set up in such a way that it can be solved exactly using known mathematical techniques. Unlike ordinary differential equations, exact differential problems have a unique solution that can be found without the need for any additional assumptions or approximations.

3. What are some common techniques used to solve exact differential problems?

Some common techniques used to solve exact differential problems include separation of variables, substitution, and integration by parts. Other techniques such as the use of integrating factors or transforming the equation into a first-order linear differential equation may also be used depending on the specific problem.

4. Can all differential equations be solved using exact methods?

No, not all differential equations can be solved using exact methods. Some equations may require the use of numerical methods or approximation techniques to find a solution. However, exact differential problems are a special class of equations that can be solved using known mathematical techniques without the need for any approximations.

5. What are some real-world applications of exact differential problems?

Exact differential problems have many applications in various scientific fields such as physics, engineering, and economics. They can be used to model and solve problems related to growth, decay, population dynamics, and many other physical and natural phenomena. They are also widely used in the development of mathematical models in fields such as finance and economics.

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