Derivative for a Galilean Tranformation

In summary, using the chain rule, we can find the values of a, b, c, and d for the given equations. After solving, it is determined that a = 1, b = 0, c = v, and d = 1.
  • #1
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Homework Statement


Using the chain rule, find a, b, c, and d:
$$\frac{\partial}{\partial x'} = a\frac{\partial}{\partial x} + b\frac{\partial}{\partial t}$$
$$\frac{\partial}{\partial t'} = c\frac{\partial}{\partial x} + d\frac{\partial}{\partial t}$$

Homework Equations


Chain rule:
$$\frac{\partial f(x,t)}{\partial x'} = \frac{\partial x}{\partial x'}\frac{\partial f}{\partial x} + \frac{\partial t}{\partial x'}\frac{\partial f}{\partial t}$$
The same form can be used for t'.

Gaililean tranformation
$$x' = x - vt$$
$$t' = t$$

The Attempt at a Solution


For x',
$$a = \frac{\partial x}{\partial x'} = \frac{\partial}{\partial x'} (x' + vt) = 1$$
$$b = \frac{\partial t}{\partial x'} = \frac{\partial}{\partial x'} (\frac{1}{v}(x-x')) = -\frac{1}{v}$$

Similarly, for t',
$$c = \frac{\partial x}{\partial t'} = \frac{\partial}{\partial t'} (x' + vt) = \frac{\partial}{\partial t'} (x' + vt')= v$$
$$d = \frac{\partial t}{\partial t'} = \frac{\partial t'}{\partial t'} = 1$$
 
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  • #2
What is your question?
 
  • #3
eys_physics said:
What is your question?
I do not know what is wrong. When I submitted my solution, it was wrong.

I am also confused. Since if t' = t, then shouldn't the partials be equivalent? That would mean c=0 and d=1.
 
Last edited:
  • #4
You have that ##t=t'##, so ##t## doesn't have any dependence on ##x'##. Therefore,
$$b=0$$
In your derivation of ##b## you are missing that ##x## depends on ##x'##.
 
  • #5
eys_physics said:
You have that ##t=t'##, so ##t## doesn't have any dependence on ##x'##. Therefore,
$$b=0$$
In your derivation of ##b## you are missing that ##x## depends on ##x'##.
So overall,
a = 1,
b= 0,
c = v,
d = 1?
 
  • #6
Yes, it should be correct.
 

1. What is a Galilean transformation?

A Galilean transformation is a mathematical model used to describe the relationship between two reference frames in relative motion. It was developed by Galileo Galilei and is based on the principles of classical mechanics.

2. How is a Galilean transformation different from a Lorentz transformation?

A Galilean transformation assumes that time and space are absolute and do not change with relative motion, while a Lorentz transformation takes into account the effects of time dilation and length contraction in special relativity.

3. What is the derivative for a Galilean transformation?

The derivative for a Galilean transformation is a mathematical operation used to calculate the rate of change of a variable with respect to another variable. In the context of Galilean transformations, it is used to describe how the position and velocity of an object change between two reference frames in relative motion.

4. How is the derivative for a Galilean transformation calculated?

The derivative for a Galilean transformation can be calculated using standard calculus techniques, such as the chain rule and product rule. It involves taking the limit as the change in time or space approaches zero, and can be expressed in terms of partial derivatives.

5. What are the practical applications of the derivative for a Galilean transformation?

The derivative for a Galilean transformation has various applications in physics and engineering, such as predicting the motion of objects in different reference frames, analyzing the behavior of systems in relative motion, and designing mechanisms for converting between different reference frames.

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