Curve of intersection of surfaces problem (Answer included).

In summary: I got:T = -14 i + 54 j + 8k but that's not the answer. It's parallel like you said if I'm correct but I can't divide each component by... to get the final t.
  • #1
s3a
818
8

Homework Statement


"Given that near (1,1,1) the curve of intersection of the surfaces
x^4 + y^2 + z^6 - 3xyz = 0
and
xy + yz + zx - 3z^8 = 0

has the parametric equations x = f(t), y = g(t), z = t with f, g, differentiable.

(a) What are the derivatives f'(1), g'(1)?

(b) What is the tangent line to the curve of intersection (1,1,1) in the forms
x = 1 + ___ s, y = 1 + ___ s, z = 1+s"


Homework Equations


Formula for gradient.


The Attempt at a Solution


For the last part, I think I see that t = 1 + s but I'm not sure.

For the first part, I computed the gradients and tried (and failed) to equate them since the surfaces are intersecting.

gradF(x,y,z) = (4x^3 + 3yz) i + (2y - 3xz) j + (6z^5 - 3xy) k
gradG(x,y,z) = (y+z) i + (x+z) j + (y+x - 24z^7) k

The correct answers are: f'(1) = 4, g'(1) = 7, x = 1 + 4s, y = 1 + 7s.

Any help in solving this problem would be greatly appreciated!
Thanks in advance!
 
Physics news on Phys.org
  • #2
You don't want to set the gradients equal. If you take their cross product you'll get a tangent vector to the intersection curve. Any idea why? And how can you use that to solve your problem?
 
  • #3
I know that taking the cross product will give a vector perpendicular to each gradient vector and I can picture it in my mind geometrically. As for how that will help me solve my problem, I'm guessing I should take the x,y, and z components and make them functions of t and then plug in t = 1 but I don't know how to proceed after the cross product step (algebraically speaking).

Here is the vector obtained from the cross product:
[(2y - 3xz)(y+x - 24z^7) - (x+z)(y+x - 24z^7)]i - [(4x^3 + 3yz)(y+x - 24z^7) - (6z^5 - 3xy)(y+z)]j + [(4x^3 + 3yz)(x+z) - (2y - 3xz)(y+z)]k
 
  • #4
s3a said:
I know that taking the cross product will give a vector perpendicular to each gradient vector and I can picture it in my mind geometrically. As for how that will help me solve my problem, I'm guessing I should take the x,y, and z components and make them functions of t and then plug in t = 1 but I don't know how to proceed after the cross product step (algebraically speaking).

Here is the vector obtained from the cross product:
[(2y - 3xz)(y+x - 24z^7) - (x+z)(y+x - 24z^7)]i - [(4x^3 + 3yz)(y+x - 24z^7) - (6z^5 - 3xy)(y+z)]j + [(4x^3 + 3yz)(x+z) - (2y - 3xz)(y+z)]k

Put x=1, y=1 and z=1 in before you take the cross product. That will make your life much easier!
 
  • #5
Oh, right. Lol!

so it's: 16 i + 160 j + 16 k ?

z = t, so t = 16 = x, so y = 10t. But these aren't correct. What am I doing wrong?
 
Last edited:
  • #6
s3a said:
Oh, right. Lol!

so it's: 16 i + 160 j + 16 k ?

That's not what I get. What are the two gradients at (1,1,1)? I've got to confess I didn't check your gradients. And I think there are some problems there. Like shouldn't the i component of grad(F) be 4x^3 - 3yz?
 
  • #7
You're right but do I have another mistake?

I now get the cross product as:
16 i + 28j + 4k
 
  • #8
s3a said:
You're right but do I have another mistake?

I now get the cross product as:
16 i + 28j + 4k

Ok, that's what I get. So stop worrying about that part. Any thoughts on where to go from there? That's a tangent to the intersection curve.
 
  • #9
I had an idea but it seems wrong.

My idea was:
I have the tangent vector to the curve. I'm assuming the curve has one variable when (being t) so having the tangent is having the derivative or something like that such that x = f'(t) = 16 but that's not the case.
 
  • #10
s3a said:
I had an idea but it seems wrong.

My idea was:
I have the tangent vector to the curve. I'm assuming the curve has one variable when (being t) so having the tangent is having the derivative or something like that such that x = f'(t) = 16 but that's not the case.

If the curve is parametrized by (f(t),g(t),t) then the tangent vector is the derivative of that at t=1. That has to be parallel to (16,28,4), it doesn't have to equal it.
 
  • #11
I have the same problem with different equations and I am getting it wrong and I checked twice and I am wondering if I got it right by fluke before or not.

Curves:
x^8 + y^4 + z^5 - 3xyz = 0
and
xy + yz + zx - 3z^4 = 0

where everything else is the same.

I got:

T = -14 i + 54 j + 8k but that's not the answer. It's parallel like you said if I'm correct but I can't divide each component by 8.
 
  • #12
s3a said:
I have the same problem with different equations and I am getting it wrong and I checked twice and I am wondering if I got it right by fluke before or not.

Curves:
x^8 + y^4 + z^5 - 3xyz = 0
and
xy + yz + zx - 3z^4 = 0

where everything else is the same.

I got:

T = -14 i + 54 j + 8k but that's not the answer. It's parallel like you said if I'm correct but I can't divide each component by 8.

You can't change the whole problem and expect to get the same answer. You have (f'(1),g'(1),1) is parallel to (16,28,4). So (f'(1),g'(1),1)=k*(16,28,4). What is k?
 
  • #13
k = 4
 
  • #14
s3a said:
k = 4

I don't think so. 4*4 isn't 1. It's close though.
 
Last edited:
  • #15
Oh, I saw the k multiplying the other vector.

k = 1/4 ?
 
  • #16
s3a said:
Oh, I saw the k multiplying the other vector.

k = 1/4 ?

Yes, k=1/4. So f'(1) and g'(1) are?
 
  • #17
4 and 7. I had successfully gotten that though ( thanks to you so thank you :) ) but for the same problem with different equations, I can't shrink the x,y and z components such that the z component is 1.
 
  • #18
s3a said:
4 and 7. I had successfully gotten that though ( thanks to you so thank you :) ) but for the same problem with different equations, I can't shrink the x,y and z components such that the z component is 1.

Different equations are different problems and have different answers. I don't know what you are asking now.
 
  • #19
I now solved it. I took the gradients and found the tangent vector from the cross product to be:

T = -14 i + 54 j + 8k

and

then I should of divided each component by 8 but being in the sleepy state I am in, I thought I had to have each component be an integer which is obviously not the case.

Thanks for all your help!
 

1. What is the definition of "Curve of intersection of surfaces problem"?

The curve of intersection of surfaces problem is a mathematical problem that involves finding the curve that results from the intersection of two or more surfaces in a three-dimensional space. This curve can be represented by a set of parametric equations or as a graph in the x-y plane.

2. How do you solve a curve of intersection of surfaces problem?

The process of solving a curve of intersection of surfaces problem involves finding the equations of the surfaces in question, setting them equal to each other, and solving for the variables. This will result in a set of parametric equations that represent the curve of intersection. These equations can be graphed or further manipulated to find specific points on the curve.

3. What are some real-world applications of curve of intersection of surfaces problems?

Curve of intersection of surfaces problems have many real-world applications, such as in engineering and architecture, where they are used to model and design structures. They are also used in computer graphics to create 3D objects and in physics to calculate the path of particles in a magnetic field.

4. What are some common challenges in solving curve of intersection of surfaces problems?

One of the main challenges in solving curve of intersection of surfaces problems is that the equations for the surfaces involved can be complex and difficult to manipulate. Additionally, the solutions may involve multiple steps and require advanced mathematical techniques. It is also important to carefully consider the domain of the parametric equations to ensure that the resulting curve is accurate.

5. Are there any software programs or tools that can assist in solving curve of intersection of surfaces problems?

Yes, there are several software programs and online tools that can assist in solving curve of intersection of surfaces problems. These include computer-aided design (CAD) software, 3D modeling programs, and online graphing calculators. These tools can help with visualizing the surfaces and their intersection, as well as performing calculations and solving equations.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
346
  • Calculus and Beyond Homework Help
Replies
3
Views
983
  • Calculus and Beyond Homework Help
Replies
3
Views
870
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
146
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
21
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
985
Back
Top