Two important points are missing in the whole presentation: 1. u =constant and v=constant give you two different ways of generating(looking at) the surface. For instance you can think of cone being made up of circles of increasing radii put one on top of the other or you can think of cone as being made up of lines passing through the origin and moving along a circle. In short a surface is made up of curves and especially u=const and v= const are called the parameter curves of the surface. 2. Surface is two dimensional object and hence will require exactly 2 parameters to describe it completely. Just as curve is one dimensional and requires only one parameter and exactly the same way if you have a solid which is 3 dimensional you will need 3 parameters ! 3. Parametrization is very useful in the sense that you could look at the object in any dimensional space and parameter will remain the same. The function r that you are talking of is actually the position vector of any point on the surface. Keeping that in mind helps too.
One thing is worth mentioning here , the point when you relate orthogonal trajectory , line and circle , is mind blowing ! 2 ) Ans : r(u,v) = (u , v , 2u+3v-6) where u : 0 to 3 , v : 0 to 4
1. u fixed: circles, v fixed: semi-circles. 2. r(u,v)=(u,v,2u+3v-6), u:[0,3], v:[0,4]. 3. u fixed: hyperbolas, v fixed: parabolas. 4. u fixed: ellipses, v fixed: parabolas. 5. u fixed: helixes, v fixed: lines. ???
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Two important points are missing in the whole presentation:
1. u =constant and v=constant give you two different ways of generating(looking at) the surface.
For instance you can think of cone being made up of circles of increasing radii put one on top of the other or you can think of cone as being made up of lines passing through the origin and moving along a circle.
In short a surface is made up of curves and especially u=const and v= const are called the parameter curves of the surface.
2. Surface is two dimensional object and hence will require exactly 2 parameters to describe it completely. Just as curve is one dimensional and requires only one parameter and exactly the same way if you have a solid which is 3 dimensional you will need 3 parameters !
3. Parametrization is very useful in the sense that you could look at the object in any dimensional space and parameter will remain the same. The function r that you are talking of is actually the position vector of any point on the surface. Keeping that in mind helps too.
Great observations madam. Thanks for sharing 😊
In second point:: small typo-- which is 4 dimensional you will need 3 parameters.
@@DrMathaholic No that's the point. Even if it is a two dimensional surface in four dim space still you will need two parameters.
@Charusheela Deshpande ohkay, that way! I thought you want to say that a surface whose dimension is 3 in R^4 then for that domain of r will be in R^3
Return of the King
Best on TH-cam for higher mathematics
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Thanks sir, we all were waiting for this series
I was also waiting to put them. Hope these series will help all :)
One thing is worth mentioning here , the point when you relate orthogonal trajectory , line and circle , is mind blowing !
2 ) Ans : r(u,v) = (u , v , 2u+3v-6) where u : 0 to 3 , v : 0 to 4
Thanks Sagar for observing and mentioning that small and important point. Also for the answer
how did you find the parameters ?
is this a paraboloid or a parabolic cylinder?
1. u fixed: circles, v fixed: semi-circles.
2. r(u,v)=(u,v,2u+3v-6), u:[0,3], v:[0,4].
3. u fixed: hyperbolas, v fixed: parabolas.
4. u fixed: ellipses, v fixed: parabolas.
5. u fixed: helixes, v fixed: lines. ???
Good work 👏 🙌