An attempt to find the total differential of a two-variable function

In summary: So, in summary, the conversation discusses the change in the function z, given by ##\Delta z##, and how it can be written in different forms. It also mentions the limit of the change as well as general paths.
  • #1
Kashmir
468
74
Let ##\quad z=h(x, y)##
and
##x=f(t) ; y=g(t)##

Let the change in the function z be given by ##\Delta z=h(x+\Delta x, y+\Delta y)-h(x,y)##

We can also write the change as

##\begin{aligned} \Delta z=h &(x+\Delta x, y)-\\ & h(x, y)-h(x+\Delta x, y) \\ &+h(x+\Delta x, y+\Delta y) \end{aligned}####\Delta z=\Delta h_{y\text { constant }}+\Delta h_{x
\operatorname{constant} }##

In the limit then we have
##dz=\frac{\partial h}{\partial x} d x+\frac{\partial h}{\partial y} d y##

Is there anything wrong with this derivation?
 
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  • #2
Sounds good.
 
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  • #3
Kashmir said:
Is there anything wrong with this derivation?
What could possibly be wrong ? :rolleyes:

$$
\Delta z=\Delta h_{y\text { constant }}+\Delta h_{x

\operatorname{constant} }$$is not really clear
 
  • #4
I've not seen this derivation, thought maybe it was wrong somehow.
BvU said:
What could possibly be wrong ? :rolleyes:

$$
\Delta z=\Delta h_{y\text { constant }}+\Delta h_{x

\operatorname{constant} }$$is not really clear
The delta y constant means that y has been kept as a constant
 
  • #5
Some amendment
[tex]\triangle z = \triangle h_{y\ constant}+\triangle h_{x+\triangle x \ constant}[/tex]
[tex]\triangle z = \triangle h_{x\ constant}+\triangle h_{y+\triangle y \ constant}[/tex]
For general paths
[tex]\triangle z = \int _{(x,y)}^{(x+\triangle x,y+\triangle y)} \nabla h \cdot \mathbf{dl}[/tex]
 
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  • #6
anuttarasammyak said:
Some amendment
[tex]\triangle z = \triangle h_{y\ constant}+\triangle h_{x+\triangle x \ constant}[/tex]
[tex]\triangle z = \triangle h_{x\ constant}+\triangle h_{y+\triangle y \ constant}[/tex]
For general paths
[tex]\triangle z = \int _{(x,y)}^{(x+\triangle x,y+\triangle y)} \nabla h \cdot \mathbf{dl}[/tex]
So am i wrong, friend?
 
  • #7
The difference is magnitude of second order of infinitesimals so
Kashmir said:
In the limit then we have
dz=∂h∂xdx+∂h∂ydy
holds in the limit.
 
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Related to An attempt to find the total differential of a two-variable function

1. What is the total differential of a two-variable function?

The total differential of a two-variable function is a mathematical concept used to describe the change in the function's output as both variables change simultaneously. It takes into account the partial derivatives of the function with respect to each variable.

2. Why is finding the total differential important?

Finding the total differential allows us to understand how a function changes in response to changes in multiple variables. This is useful in many fields, including physics, economics, and engineering, where multiple variables often affect the outcome of a system.

3. How do you find the total differential of a two-variable function?

The total differential can be found by taking the partial derivative of the function with respect to each variable and multiplying it by the corresponding change in that variable. These values are then added together to give the total differential.

4. Can the total differential be used to approximate the change in a function?

Yes, the total differential can be used as an approximation for the change in a function when the changes in the variables are small. This is known as the total differential approximation or the linear approximation.

5. Is the total differential the same as the total derivative?

No, the total differential and the total derivative are different concepts. The total derivative is a generalization of the derivative to functions with multiple variables, while the total differential is a specific calculation used to describe changes in a two-variable function.

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