Surfaces in Space / Vector-Valued Functions

In summary, the conversation discusses sketching a space curve formed by the intersection of two surfaces and representing it as a vector-valued function. The surfaces are z=x2+y2 and x+y=0, and the parameter is x=t. The final answer is r(t)=<t,-t,2t2>.
  • #1
llauren84
44
0

Homework Statement



Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function.


Homework Equations



Surfaces: z=x2+y2, x+y=0
Parameter: x=t


The Attempt at a Solution



So, I sketched the space curve represented by the intersection of the two curves. It's simply a parabola. However, I can't seem to think of how to put it into a vector-valued function. I'm getting a mind block. (Perhaps, before anyone answers this, I may come up with the answer.)
 
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  • #2
Got it yet? You already have x=t. What's y in terms of t? (Put x=t into the plane equation). Finally z=?
 
  • #3
So, it's just z=t2+(-t)2? and..no I didn't get it yet. I put down my math and did other things. Thanks =) So, it's r(t)=<t,-t,2t2>?
 
Last edited:
  • #4
Yes, that's z.
 

Related to Surfaces in Space / Vector-Valued Functions

1. What are surfaces in space?

Surfaces in space refer to any two-dimensional figure or object that exists in a three-dimensional space. This can include objects such as spheres, cubes, cones, and more complex surfaces like a torus or a hyperboloid.

2. How are surfaces in space represented mathematically?

Surfaces in space are often represented using vector-valued functions, which map a two-dimensional input (usually denoted by u and v) to a three-dimensional output (x, y, z). This allows for a precise and mathematical description of the surface's shape and position in space.

3. What is the significance of surfaces in space in science and engineering?

Surfaces in space play a crucial role in many scientific and engineering fields, including physics, astronomy, computer graphics, and more. They are used to model and study various phenomena, such as planetary surfaces, fluid flow, and electromagnetic fields.

4. Can surfaces in space be described using other mathematical concepts?

Yes, surfaces in space can also be described using other mathematical concepts such as parametric equations, implicit equations, and level sets. However, vector-valued functions are often the most convenient and precise way to represent and manipulate surfaces in space.

5. How are surfaces in space visualized and studied?

There are various tools and techniques used to visualize and study surfaces in space, such as computer software, physical models, and mathematical analysis. These methods allow scientists and engineers to better understand the properties and behavior of these surfaces and how they interact with their surroundings.

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