Real Analysis with Physics

In summary: Hence the LHS is equal to:##\int_{x_1}^{x_2} \vec{F} \cdot dx = \frac12 m \int_{t_1}^{t_2} \frac{d}{dt}(v^2(t)) dt = \frac12 m (v^2(t_2) - v^2(t_1))##which is the same as the RHS.
  • #1
aliens123
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Suppose I wanted to prove the work-kinetic energy theorem. This means that I want to show that
[itex]\frac{1}{2}m( \vec {v}^2_f - \vec{v}^2_i)=\int_{x_1}^{x_2} \vec{F} \cdot dx[/itex].

So, I go ahead and start on the right side:

[itex]\int_{x_1}^{x_2} (m \frac{d\vec{v}}{dt}) \cdot dx = m \int_{x_1}^{x_2} (\frac{d\vec{x}}{dt}) \cdot dv=m \int_{x_1}^{x_2} \vec{v} \cdot dv=\frac{1}{2}m( \vec {v}^2_f - \vec{v}^2_i)[/itex].

And I say that I am done. But my question is, how do we rigorously argue that the following step is valid?:
[itex]\int_{x_1}^{x_2} (m \frac{d\vec{v}}{dt}) \cdot dx = m \int_{x_1}^{x_2} (\frac{d\vec{x}}{dt}) \cdot dv[/itex]

In other words, if we were in a real analysis class, what would allow us to switch the [itex]d\vec{v}[/itex] with the [itex]d\vec{x}[/itex], using just the formal definition of an integral? Intuitively if we think of these as representing infinitesimally small amounts which are multiplied, then obviously the multiplication is commutative. But this is not very satisfying. What role does the [itex]d\vec{x}[/itex] actually play in the integral?
 
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  • #2
Doesn't this look like a partial integration to an analysis expert ?
[edit] never mind, just woke up.o:)o:)o:)
 
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  • #3
aliens123 said:
Suppose I wanted to prove the work-kinetic energy theorem. This means that I want to show that
[itex]\frac{1}{2}m( \vec {v}^2_f - \vec{v}^2_i)=\int_{x_1}^{x_2} \vec{F} \cdot dx[/itex].

The first thing is to define what is meant by a line integral along a curve. First you have to parameterise the curve and in this case using time ##t## is the best option. By definition:

##\int_{C} \vec{F} \cdot \vec{dr} = \int_{t_1}^{t_2} \vec{F}(\vec{r(t)}) \cdot \vec{r'(t)} dt##

Where ##\vec{r(t)}## is a parameterisation of the curve ##C##.

In this case we have:

##\vec{F}(\vec{r(t)}) \cdot \vec{r'(t)} = m \vec{r''(t)}\cdot \vec{r'(t)} = m(\frac12) \frac{d}{dt}(\vec{r'(t)} \cdot \vec{r'(t)}) = m(\frac12) \frac{d}{dt}(v^2(t))##

Hence:

##\int_{C} \vec{F} \cdot \vec{dr} = \frac12 m \int_{t_1}^{t_2} \frac{d}{dt}(v^2(t)) dt = \frac12 m (v^2(t_2) - v^2(t_1))##

The last integral is just an ordinary integral wrt ##t## and we can apply the fundamental theorem of calculus.
 
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1. What is the purpose of studying Real Analysis with Physics?

The purpose of studying Real Analysis with Physics is to understand the mathematical foundations and principles that govern physical systems. Real Analysis provides the tools and techniques to analyze and solve complex problems in physics, while also deepening our understanding of the underlying mathematical concepts.

2. What are some key concepts in Real Analysis that are relevant to physics?

Some key concepts in Real Analysis that are relevant to physics include limits, continuity, differentiation, integration, and series. These concepts are essential in understanding the behavior of physical systems and form the basis for many mathematical models used in physics.

3. How is Real Analysis used in practical applications of physics?

Real Analysis is used in practical applications of physics in a variety of ways. For example, it is used to derive equations that describe the motion of objects, calculate the forces acting on a system, and analyze the behavior of waves and electromagnetic fields. Real Analysis also plays a crucial role in developing and testing theories in physics.

4. Can Real Analysis be applied to any branch of physics?

Yes, Real Analysis can be applied to any branch of physics. Its principles and techniques are used in classical mechanics, electromagnetism, thermodynamics, quantum mechanics, and many other areas of physics. Real Analysis provides a universal framework for understanding and solving problems in the physical sciences.

5. Are there any real-world examples that demonstrate the importance of Real Analysis in physics?

Yes, there are many real-world examples that demonstrate the importance of Real Analysis in physics. For instance, the laws of motion developed by Isaac Newton and the theory of relativity proposed by Albert Einstein both rely heavily on Real Analysis concepts. In addition, the study of fluid dynamics, which is crucial in fields such as aerodynamics and oceanography, heavily utilizes Real Analysis techniques.

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