Finding a tangent vector to the intersection of two surfaces

In summary, the problem asks to find a tangent vector to the intersection of two surfaces, S1: z = x^2 + y^2 and S2: x^2 + y^2 = 2x + 2y, at the point (0, 2, 4). There are two different proposed solutions, one involving finding gradients and cross products, and the other involving finding a vector orthogonal to the function z = 2x + 2y. There is also confusion about whether the given surfaces are actually planes, and whether there can be infinite tangential vectors to the intersection.
  • #1
hangainlover
83
0

Homework Statement


The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y
intersect at a curve gamma
. Find a tangent vector to
at the point (0, 2, 4).


Homework Equations


i thought about finding gradients of the two functions and plug in the given point in the gradients and cross product the two.
I get (2,2,8) as the vector tangent to the intersection gamma.

My solution was saying " If we produce two normal vectors n1 ? S1 and n2 ? S2, any
vector perpendicular to n1 and n2 (specifically n1 × n2) would be tangent to
both S1 and S2 and thus tangent to their common intersection "
.
it didnt make sense to me.
Cause if the vector is tangent to s2, x2 + y2 = 2x + 2y, then the vector should have zero for z component. the intersection is going to be some kinda curve , so i think there will be infinite number of vectors tangential to the intersectino between s1 and s2. and the cross product of the two gradients is just one of them.
I don't think the vector we get by cross producting the two gradients has to be tangential to s1, and s2


The Attempt at a Solution

 
Physics news on Phys.org
  • #2
i also noted the the intersection is part of z=2x+2y
so why not just find a vector orthogonal to the function z=2x+2y
'?
can't any vector be tangential to a curve? i suppose we can just write a random vector and say it is tangential to the given intersection.
im confused
 
  • #3
in general, i know you need to find two gradients and cross them to get the vector tangential to the intersection of the two.
but in this case, since the intersection is just a plane, i thought there can be infinite number of vectors tangential to the intersection.
 
  • #4
hangainlover said:

Homework Statement


The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y
intersect at a curve gamma
. Find a tangent vector to
at the point (0, 2, 4).
Do x2 and y2 happen to mean x2 and y2? If so, at the very least write these as x^2 and y^2 to give readers the tiniest clue as to what you are talking about.
 

1. What is the definition of a tangent vector?

A tangent vector is a vector that is tangent to a curve or surface at a given point. It represents the direction and rate of change of the curve or surface at that point.

2. How do you find the intersection of two surfaces?

To find the intersection of two surfaces, you must set the equations of the two surfaces equal to each other and solve for the variables. The resulting solution will give the coordinates of the points where the two surfaces intersect.

3. Why is finding a tangent vector to the intersection of two surfaces important?

Finding a tangent vector to the intersection of two surfaces is important because it allows us to understand the behavior of the surfaces at the point of intersection. This information can be useful in various applications, such as optimizing surfaces for specific purposes or studying the geometry of the surfaces.

4. What is the process for finding a tangent vector to the intersection of two surfaces?

The process for finding a tangent vector to the intersection of two surfaces involves finding the partial derivatives of both surfaces with respect to each variable, setting them equal to each other, and solving for the variables. The resulting solution will give the direction and rate of change of the surfaces at the point of intersection, which can be represented by a tangent vector.

5. Can a surface have more than one tangent vector at a point of intersection?

Yes, a surface can have more than one tangent vector at a point of intersection. This is because a surface can have multiple curves or lines passing through a single point, each with its own tangent vector. However, there will always be one unique tangent vector that best represents the overall behavior of the surfaces at that point of intersection.

Similar threads

Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
466
Tangent vector on the intersection of surfaces
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Find the vector equation of the line that passes through the point P and intersects with the straight lines R and S
    • Calculus and Beyond Homework Help
Replies
5
Views
946
Find all points of intersection
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Find all points where surface normal is perpendicular to plane
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Surface integral: Calculate the heat flow from a cylinder
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Problem about tangent plane to surface
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Find all points where the level surface tangent plane is parallel
    • Calculus and Beyond Homework Help
Replies
4
Views
3K
Create curve function from intersection of two surfaces
    • Calculus and Beyond Homework Help
Replies
4
Views
923
Unit Vector Problem: Find Point of Intersection
    • Calculus and Beyond Homework Help
Replies
2
Views
3K

Hot Threads

Recent Insights

Back
Top