I Help with understanding inexact differential

Tags:
1. Nov 13, 2017

granzer

In

How is equation 145 giving a direction(ie gradient) and not a slope?.

Also here

how is equation 147 arrived at?

Any help would be much appreciated.

2. Nov 13, 2017

granzer

Can someone please explain this to me. Haven't been able to find an answer .

3. Nov 13, 2017

Staff: Mentor

I can just barely read the images you posted.
For your first question, in the US, we tend to call $\frac {dy}{dx}$ a slope; in Europe, people tend to call this a gradient. I prefer to reserve the term gradient to functions such as this: If z = f(x, y), then $\nabla z = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$

If y = 3x, then y' or ($\frac{dy}{dx}$) = 3. This gives a direction in the sense that from any point on the graph of this line, you can get to another point by going right 1 unit and then up 3 units.

Same idea for your second question.

4. Nov 14, 2017

granzer

@Mark44 Hello Sir,

Thank you so much. The answers u gave have cleared my doubt about equation 147 and I completely agree with you. I was wondering if it should be given as a slope. And yes going right 1 step and going up 3 step gives a direction from the point. But it is also true going 3 step down and one step back we can get another point. So the direction is both front and back and not a particular direction that would be given by a gradient. But if we take gradient to mean slope here then my doubt is cleared.

But with regard to eq 147, it is given:

$∂σ/∂x=X'∂σ/∂y=X'Y'/τ$

I am not understanding how #X′Y′/τ# is got here.
Any clarification on this would be really helpful.
So sorry about the image quality. The image I uploaded was very good. Don't know how to upload the image with that resolution here. Here is the image I uploaded https://i.stack.imgur.com/APHR9.png

Also, the first image is https://i.stack.imgur.com/cj7YG.png