Meaning of \delta in Implicit Functions

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In summary, the conversation discusses the use of implicit functions and the calculation of their derivatives. The use of \delta and ∂ in the equations is explained, with the understanding that \delta represents a small change and ∂ represents a partial derivative. The concept of taking differentials and the meaning of the term dx is also clarified.
  • #1
Moonspex
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I've come across a very ambiguous statement in my notes on implicit functions (part of the partial derivatives part of the course). I'll write out the preceding explanation but the problematic line is marked by *

"Sometimes we can define a function z=z(x, y) only in implicit form, i.e. through an equation F(x, y, z) = 0.
It is not always possible to solve this equation for z and obtain the function z=f(x,y).

In order to calculate the derivatives of a function defined implicitly we note that from the above equation it follows that:
* [tex]\delta[/tex]F=0 [tex]\Rightarrow[/tex] [tex]\delta[/tex]F = Fx[tex]\delta[/tex]x + Fy[tex]\delta[/tex]y + Fz[tex]\delta[/tex]z = 0.
Or by taking differentials,
Fxdx + Fydy + Fzdz = 0"

My main problem is understanding how [tex]\delta[/tex]x can stand on its own (above used as a factor). Is it just the same as ∆x, i.e. a change in x and not a derivative?

Also, how the [tex]\delta[/tex] expressions change to d expressions in the second line is unclear to me...
 
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  • #2
There is some ambiguity in the meaning of delta. Ask a physicist and he/she will tell you that [tex]\delta[/tex] means a small change whereas [tex]\Delta[/tex] corresponds to a large change.
 
  • #3
i believe that your usage of ∂ is in the form of a partial derivative
and that you are using partial differentials in order to gain a total differential of the function F(x,y,z)
 
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  • #4
raymo39 said:
i believe that your usage of ∂ is in the form of a partial derivative
and that you are using partial differentials in order to gain a total differential of the function F(x,y,z)
The problem is that the Fx term implies ∂F/∂x and so on, whereas these individual partial derivative are then multiplied by the [tex]\delta[/tex]x and so on terms - this is what confuses me.

So is it a partial derivative multiplied by a small change?
 
  • #5
Yes, [itex]\delta x[/itex] here just means a small change in x. Taking the limit as [itex]\delta x[/itex] becomes "infinitesmal" gives you dx.
 
  • #6
Ok, so what does dx stand for then? It is again multiplied by Fx, giving ∂F/∂x · dx. Since it isn't a double derivative of F with respect to x (especially since ∂ =/= d), then what is it's meaning here?
 
  • #7
dx is a "differential".
 
  • #8
So is it similar to "an infinitesimal change in x" but now it's an "infinitesimally small difference (ie, variation or difference) in x"?
 
  • #9
well when you are taking differentials, you add limits to your small change in the value as they tend to zero
 

1. What does delta represent in implicit functions?

Delta is a symbol that is commonly used in mathematics to represent a small change or difference in a variable. In implicit functions, delta typically refers to a small change in the independent variable, which affects the dependent variable.

2. How is delta used in implicit differentiation?

In implicit differentiation, delta is used to represent a small change in the independent variable, which allows us to find the derivative of the dependent variable with respect to the independent variable. This is useful when the dependent variable cannot be easily expressed in terms of the independent variable.

3. Can delta have a negative value in implicit functions?

Yes, delta can have a negative value in implicit functions. This indicates a decrease or decrease in the independent variable, which will result in a corresponding change in the dependent variable.

4. How do you calculate delta in implicit functions?

To calculate delta in implicit functions, you need to find the difference between two values of the independent variable. This can be done by subtracting the initial value from the final value. The resulting difference represents the change in the independent variable.

5. Why is delta important in implicit functions?

Delta is important in implicit functions because it helps us understand how a small change in the independent variable can affect the dependent variable. This is useful in analyzing the behavior of functions and finding derivatives, especially when the functions are not easily expressed in terms of the independent variable.

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