Interchanging partial derivatives and integrals

In summary, the conversation revolved around the question of whether partial differentiation and partial integration commute with each other, and if not, what is the relationship between the two operations. The fundamental theorem of calculus was mentioned as a possible analogy, and the formula for the expression \frac{\partial}{\partial t}\int^{t}_{0}F(t, \tau) \partial \tau was discussed. The use of the Heaviside step function and its derivative as a delta function were also mentioned.
  • #1
lugita15
1,554
15
In the midst of https://www.physicsforums.com/showthread.php?t=403002", I came upon a rather interesting, though probably elementary, question. Analagous to the fundamental theorem of calculus, is there a formula or theorem concerning the expression [tex] \frac{\partial}{\partial t}\int^{t}_{0}F(t, \tau) \partial \tau [/tex]. In other words, does partial differentiation with respect to one variable commute with partial integration with respect to the other variable? If they don't commute, what is the relation between the two operations?

The fundamental theorem of calculus (well, the first part anyway) came from the non-rigorous intuition that [tex]\int^{x+dx}_{x}f(t)dt=f(x)dx[/tex]. Can a similar intuitive statement be made about [tex]\int^{t+dt}_{t}F(t, \tau) \partial \tau[/tex]? We know it has to be proportional to [tex]dt[/tex], but what is the constant of proportionality?
 
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  • #2
If you don't want to use the Heaviside step function, as I suggested in the other thread, you might prefer this
 
  • #3
gabbagabbahey said:
If you don't want to use the Heaviside step function, as I suggested in the other thread, you might prefer this
Thanks. That's exactly the kind of thing I wanted. Using the formula in the end of the subsection entitled "General form with variable limits," I get
[tex]\frac{\partial}{\partial t}\int^{t}_{0}F(t, \tau) \partial \tau = F(t,t)+\int^{t}_{0}\frac{\partial F}{\partial t} \partial \tau[/tex]
 
  • #4
That's correct; provided [itex]F[/itex] is piecewise smooth everywhere, exponentially bounded, and continuous at [itex]\tau=t[/itex] . You'd get the same thing using the Heaviside step function (really a distribution or generalized function, not an ordinary function) since its derivative is a delta function.
 
  • #5
Based on this formula, in the other thread I just tried to write an expression for the derivative of a convolution. I hope I'm right.
 

1. What is the concept behind interchanging partial derivatives and integrals?

The concept behind interchanging partial derivatives and integrals is known as the Fundamental Theorem of Calculus. It states that the derivative and integral of a function are inverses of each other, allowing for the interchange of the two operations.

2. When can you interchange partial derivatives and integrals?

Partial derivatives and integrals can be interchanged in cases where the function is continuous and has a well-defined partial derivative. Additionally, the limits of integration must be constants rather than variables.

3. What is the notation used for interchanging partial derivatives and integrals?

The notation used for interchanging partial derivatives and integrals is known as the Leibniz notation. It involves placing the integral sign and limits of integration on the same line as the partial derivative symbol.

4. Can partial derivatives and integrals be interchanged for any function?

No, partial derivatives and integrals cannot be interchanged for any function. The function must satisfy certain criteria, such as being continuous and having a well-defined partial derivative.

5. What are some real-world applications of interchanging partial derivatives and integrals?

Interchanging partial derivatives and integrals is commonly used in physics and engineering to calculate quantities such as work, potential energy, and electric fields. It is also used in economics to analyze production functions and marginal cost.

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