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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 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.
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.