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I Help with understanding inexact differential

  1. Nov 13, 2017 #1
    In

    vRe9k.png

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

    Also here

    rexHO.png

    how is equation 147 arrived at?

    Any help would be much appreciated.
     
  2. jcsd
  3. Nov 13, 2017 #2
    Can someone please explain this to me. Haven't been able to find an answer .
     
  4. Nov 13, 2017 #3

    Mark44

    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.
     
  5. Nov 14, 2017 #4
    @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
     
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