Can function transformation result in a constant variation?

In summary: This would imply that the Lagrangian is a scalar, which it isn't.Yes, that's how I tend to write things, although most books seem to prefer to put a prime on the Lagrangian as well. I would write:##f'(x) = f(x') = f(g(x))##, assuming ##x' = g(x)##But, some texts write ##f'(x') = f'(x)##, which (technically) I think is inaccurate. This would imply that the Lagrangian is a scalar, which it isn't.In summary, the conversation discusses the transformation of a scalar function and its coordinate transformation, resulting in a variation of the function. The confusion arises due
  • #1
Higgsono
93
4
Given a scalar function, we consider the following transformation:

$$\delta f(x) = f'(x') - f(x) $$ Given a coordinate transformation $$x' = g(x)$$

But since ##f(x)## is a scalar isn't it true that ##f'(x') = f(x) ##

Then the variation is always zero? What am I missing?
 
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  • #2
Higgsono said:
Given a scalar function, we consider the following transformation:

$$\delta f(x) = f'(x') - f(x) $$

But since ##f(x)## is a scalar isn't it true that ##f'(x') = f(x) ##

Then the variation is always zero? What am I missing?

Do you have more context?
 
  • #3
PeroK said:
Do you have more context?
PeroK said:
Do you have more context?

Given a coordinate transformation $$x' = g(x)$$
 
  • #4
and ##f'##?
 
  • #5
PeroK said:
and ##f'##?

I don't know. But they always write a prime on the function as well.
 
  • #6
Higgsono said:
I don't know. But they always write a prime on the function as well.

If you don't know what it is, how can you calculate ##\delta f(x)##.

My initial interpretation of ##f'## would agree with what you said in your OP. That ##f'(x') = f(x)## by definition.
 
  • #7
PeroK said:
If you don't know what it is, how can you calculate ##\delta f(x)##.

My initial interpretation of ##f'## would agree with what you said in your OP. That ##f'(x') = f(x)## by definition.

f is a scalar field, and I am considering variations of this field in the Lagrangian. But if $$ f'(x') = f(x)$$ by definition. What is the point of varying the fields?
 
  • #8
Higgsono said:
f is a scalar field, and I am considering variations of this field in the Lagrangian. But if $$ f'(x') = f(x)$$ by definition. What is the point of varying the fields?

Ah, Lagrangian, you see, the magic word!

The Lagrangian ##f'## has the same form as the Lagrangian ##f##. It's not the function obtained by a coordinate transformation. It's the same function as ##f## applied to the coordinates ##x'##.
 
  • #9
PeroK said:
Ah, Lagrangian, you see, the magic word!

The Lagrangian ##f'## has the same form as the Lagrangian ##f##. It's not the function obtained by a coordinate transformation. It's the same function as ##f## applied to the coordinates ##x'##.

So if it has the same form, the transformation would be $$\delta f= f(x') - f(x)$$ intead of $$\delta f= f'(x') - f(x)$$

I'm always confused why they put a prime on the function when they consider coordinate transformations.
 
  • #10
Higgsono said:
So if it has the same form, the transformation would be $$\delta f= f(x') - f(x)$$ intead of $$\delta f= f'(x') - f(x)$$

Yes, that's how I tend to write things, although most books seem to prefer to put a prime on the Lagrangian as well. I would write:

##f'(x) = f(x') = f(g(x))##, assuming ##x' = g(x)##

But, some texts write ##f'(x') = f'(x)##, which (technically) I think is inaccurate.
 

1. What is the definition of a transformation of a function?

A transformation of a function refers to the process of changing or manipulating the graph of a function without changing its basic shape. This can be done by applying certain operations, such as shifting, stretching, or reflecting, to the original function.

2. How do you determine the type of transformation applied to a function?

The type of transformation applied to a function can be determined by analyzing the changes in the function's equation. For example, if the equation is modified by adding or subtracting a constant value, it indicates a vertical shift. If the equation is multiplied by a constant, it indicates a vertical stretch or compression. Changes in the variables (x and y) indicate horizontal shifts or reflections.

3. What is the difference between a horizontal and vertical transformation?

A horizontal transformation refers to changes made to the input variable (x) of a function, which affects the horizontal position of the graph. This can include shifting the graph left or right, or reflecting it across the y-axis. A vertical transformation, on the other hand, refers to changes made to the output variable (y), which affects the vertical position of the graph. This can include shifting the graph up or down, or reflecting it across the x-axis.

4. Can multiple transformations be applied to a function?

Yes, multiple transformations can be applied to a function. These transformations are applied in a specific order, known as the transformation sequence. The order in which the transformations are applied can affect the final graph of the function. The general rule is that horizontal transformations are applied first, followed by vertical transformations.

5. How can transformations be used to solve real-world problems?

Transformations of functions have many real-world applications, such as in physics, economics, and engineering. For example, in physics, transformations can be used to model the motion of objects, while in economics, they can be used to analyze supply and demand curves. In engineering, transformations can be used to design and optimize structures. By understanding how to transform functions, scientists can better understand and solve real-world problems.

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