Difference between a closed solid and a Cartesian surface

In summary, the author struggles to distinguish between a closed solid and a cartesian surface. He asks for help understanding the exercises from an exam. He is informed that z=x^2+y^2 implies that the surface is a surface.
  • #1
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Homework Statement
look at the image
Relevant Equations
-closed solid
-cartesian surface
-Divergence theorem
Greetings All!

I have hard time to make the difference between the equation of a closed solid and a cartesian surface.

For example in the exercice n of the exam I thought that the equation was describing a closed solid " a paraboloid locked by an inclined plane (so I thought I could use the divergence theorem) while it was describing a broken paraboloid (a cartesian surface). I joined the exercice to be more precise.

I hope I could explain my problem.

Thank you
exercice.png


Best regards.
 
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  • #2
You are told that ##z= x^2 + y^2## on ##\Sigma##. This is not generally true on the plane that would close the surface.
 
  • #3
Any time a set in ##R^n## has an equal sign, "=", in the definition, there is no n-dimensional volume to the set, so it is a surface (boundary?). So ##z = x^2 + y^2## implies that it is a surface.
 
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  • #4
FactChecker said:
Any time a set in ##R^n## has an equal sign, "=", in the definition, there is no n-dimensional volume to the set, so it is a surface (boundary?). So ##z = x^2 + y^2## implies that it is a surface.
thank you very much Z= something means surface !
 
  • #5
Orodruin said:
You are told that ##z= x^2 + y^2## on ##\Sigma##. This is not generally true on the plane that would close the surface.
the plane that would close the surface would have been z=2y+3
 
  • #6
Amaelle said:
the plane that would close the surface would have been z=2y+3
Note: If you had been told z=2y+3 instead of ##z\leq 2y+3##, your set would have been a one-dimensional curve (the intersection of the two surfaces). Here you are told that you have a single surface (the parabola) and that it is the part of the parabola inside a given volume (given by the inequality).
 
  • #7
Orodruin said:
Note: If you had been told z=2y+3 instead of ##z\leq 2y+3##, your set would have been a one-dimensional curve (the intersection of the two surfaces). Here you are told that you have a single surface (the parabola) and that it is the part of the parabola inside a given volume (given by the inequality).
yes thanks a million!
 

What is a closed solid?

A closed solid is a three-dimensional object that is completely enclosed and has a finite volume. It has a definite shape and is not open or hollow.

What is a Cartesian surface?

A Cartesian surface is a two-dimensional surface that is defined by a set of coordinates in a three-dimensional Cartesian coordinate system. It is a mathematical concept used in geometry and calculus.

What is the difference between a closed solid and a Cartesian surface?

The main difference between a closed solid and a Cartesian surface is that a closed solid is a physical object with volume and shape, while a Cartesian surface is a mathematical concept with no physical existence.

Can a closed solid also be a Cartesian surface?

Yes, a closed solid can also be a Cartesian surface if it is defined by a set of coordinates in a three-dimensional Cartesian coordinate system. For example, a cube can be represented as a Cartesian surface with six faces defined by six sets of coordinates.

Can a Cartesian surface be a closed solid?

No, a Cartesian surface cannot be a closed solid because it is a two-dimensional mathematical concept and does not have volume or a definite shape. It is only a representation of a three-dimensional object in a mathematical space.

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