Integrating over Triangle C: Computing Normal Vector

In summary, to compute the given integral, we separate it into three sub-integrals over the three sides of the triangle using the corresponding normal vectors. It is necessary to parameterize each side before integrating and adding them up.
  • #1
Somefantastik
230
0

Homework Statement



C is triangle (0,0), (4,0), (0,3). R is the enclosed region. Compute the following integral, where n is the outward pointing normal:

[tex] \int_{C} \left(4x-y^{2}\right)n^{1}ds [/tex]

where [tex] n^{1} = \widehat{i} \cdot \widehat{n} [/tex]

Homework Equations





The Attempt at a Solution



I can't remember how to get the normal vector, can someone start me out there?
 
Physics news on Phys.org
  • #2
There are three normal vectors, one for each side of the triangle that encloses the region. If the vector (a,b) is a tangent to a side then (-b,a) is a normal, isn't it? It's not necessarily a unit normal, but you should know how to fix that. Is that enough to get you started?
 
  • #3
So to evaluate this integral, should I separate it into 3 sub integrals over the 3 sides, using the corresponding normals?
 
  • #4
Also, is it necessary to parameterize before integrating? I'm getting hung up on the little details and missing then big picture.
 
  • #5
Yes, separate it into three integrals. Decide which direction around the triangle you are going. Then parameterize each side by length, integrate and add them up.
 

FAQ: Integrating over Triangle C: Computing Normal Vector

1. What is the purpose of integrating over Triangle C when computing the normal vector?

The purpose of integrating over Triangle C is to determine the direction and magnitude of the normal vector. This vector is perpendicular to the surface of the triangle and is used in various mathematical and physical calculations.

2. How do you find the normal vector of a triangle?

To find the normal vector of a triangle, you can use the cross product of two sides of the triangle. This will result in a vector that is perpendicular to the triangle's surface and has a magnitude equal to the area of the triangle.

3. What is the formula for computing the normal vector of a triangle?

The formula for computing the normal vector of a triangle is N = (b-a) x (c-a), where a, b, and c are the vertices of the triangle. This formula uses the cross product to find the vector perpendicular to the triangle's surface.

4. Can you integrate over any shape to compute the normal vector?

Yes, you can integrate over any shape to compute the normal vector. However, the shape must have a well-defined surface and the integration must be done using appropriate techniques. For a triangle, integration can be done using the formula N = (b-a) x (c-a).

5. Can the normal vector of a triangle be negative?

Yes, the normal vector of a triangle can be negative. This indicates that the direction of the vector is opposite to the direction of the positive normal vector. In other words, the vector is pointing in the opposite direction to the surface of the triangle.

Similar threads

Back
Top