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partial derivatives of surfaces like that of drinking glass

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PG1995

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Hi

Could you please help me with the query **broken link removed**? Thank you.

Regards
PG
 
The idea of using 3D objects like glass and vase is too confusing without more clarification.

z=f(x,y) is a 2D surface function, so you need to specify an appropriate object.

For example, I would have said, " fill the glass with water and consider the boundary between the glass and the water as a 2D surface function. "

You can take the partial derivative of such a function.
 
The idea of using 3D objects like glass and vase is too confusing without more clarification.

z=f(x,y) is a 2D surface function, so you need to specify an appropriate object.

For example, I would have said, " fill the glass with water and consider the boundary between the glass and the water as a 2D surface function. "

You can take the partial derivative of such a function.

Or, perhaps you can imagine a drinking glass which has no thickness; it's just has a defined boundary(ies). In other words, it's just a mathematical object. How would find its partial derivatives in the context of what I say **broken link removed**? Thank you for the help.

Best regards
PG
 
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Or, perhaps you can imagine a drinking glass which has no thickness; it's just has a defined boundary(ies). In other words, it's just a mathematical object. How would find its partial derivatives in the context of what I say here ...

Yes, an ultra thin glass works too. I just want to make sure your issues are not related to the 3D aspects of a solid object.

However, there is a reason why I wanted to specify the interior boundary between water and glass. Your diagram shows the inside of the glass to be a function since the walls on the inside are not vertical. The outside of the glass appears to have vertical walls which do not obey the definition of a 2D surface function (i.e. single valued).

So, if the inside of the glass is a valid 2D surface function, why can't you just take the partial derivative as specified in your document? I don't see any issue in holding one variable constant and finding the slope along the other axis. Is their description confusing, or does the glass seem to have different properties than the surface function they show? Please explain the issue more.
 
Hello,

We could easily find a function that describes the inside and outside of the glass, probably as a solid of revolution.

But i think what is important here is that derivatives dont always exist. That's just the way it is. When that happens we might find other ways to deal with the situation. For example, if we run into a vertical line we cant find a derivative dy/dx, but we might find a derivative dx/dy and that might allow us to proceed in another manner.
 
Thank you, Steve, MrAl.

Yes, an ultra thin glass works too. I just want to make sure your issues are not related to the 3D aspects of a solid object.

However, there is a reason why I wanted to specify the interior boundary between water and glass. Your diagram shows the inside of the glass to be a function since the walls on the inside are not vertical. The outside of the glass appears to have vertical walls which do not obey the definition of a 2D surface function (i.e. single valued).

So, if the inside of the glass is a valid 2D surface function, why can't you just take the partial derivative as specified in your document? I don't see any issue in holding one variable constant and finding the slope along the other axis. Is their description confusing, or does the glass seem to have different properties than the surface function they show? Please explain the issue more.

It looks like if we think of using **broken link removed** glass **broken link removed** instead, then the partial derivatives can be found. Please have a look. Thank you for the help.

Regards
PG
 
Thank you, Steve, MrAl.



It looks like if we think of using **broken link removed** glass **broken link removed** instead, then the partial derivatives can be found. Please have a look. Thank you for the help.

Regards
PG

Have you studied "conic sections" before?

http://math2.org/math/algebra/conics.htm

The glass you show is at least a little bit like part of a cone (on the sidewalls at least). So, taking your partial derivatives with respect to y is like making a conic section by dropping a vertical plane thru the cone. This gives a hyperbolic-like shape (perhaps not a perfect hyperbola, since the glass is not a perfect cone). There is no problem finding slopes/tangents on these types of curves.

Hopefully, this gives you the tool to visualize it now.
 
Have you studied "conic sections" before?

http://math2.org/math/algebra/conics.htm

The glass you show is at least a little bit like part of a cone (on the sidewalls at least). So, taking your partial derivatives with respect to y is like making a conic section by dropping a vertical plane thru the cone. This gives a hyperbolic-like shape (perhaps not a perfect hyperbola, since the glass is not a perfect cone). There is no problem finding slopes/tangents on these types of curves.

Hopefully, this gives you the tool to visualize it now.

Thank you.

I have been through the conic sections but only from elementary point of view. Please have a look on the **broken link removed** and please note that in the diagram I forgot to mention that the vertical plane passes through the "P1" and the "P1" would lie at the bottom of parabola shape intersected by the plane. Please let me know if I have it correct this time. Thank you.

Regards
PG
 
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I think you understand it now.

However, strictly, the shape is not a parabola, but more like a hyperbolic shape. If you look at the theory of conic sections, you will see that the plane cutting the cone at a skewed angle gives the parabola, while the vertical plane cutting the cone gives the hyperbola.
 
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