Calculating Surface Area on a Bounded Tetrahedron Plane

[adichy]

Homework Statement

4. The domain D is a tetrahedron bounded by the planes x = 0, y = 0, z = 0 and
x + y + z = 1 Calculate
(a) The volume of the domain.
(b) The x-coordinate of the centre-of-mass of the domain, assuming constant density.
(c) Find, in terms of x and y the vector R from the origin to a point on the plane
x + y + z = 1.
(d) Find the (vectorial) element of surface area dS on that plane, in terms of x, y, dx
and dy.
(e) Hence calculate the area of the portion of that plane on the surface of the domain
D

Homework Equations

The Attempt at a Solution

ive done a, b, and c and I am looking for some direction for solving d and e...not looking for the answer here but what exactly is the question asking me to do and what's the general method of solving them...thx for ur help

 

[LCKurtz]

If you have a surface parameterized as

\vec R(u,v) = \langle x(u,v),y(u,v),z(u,v)\rangle

the vector element of surface area is

d\vec S = \vec R_u \times \vec R_v\ dudv

In your example, you might use x and y as the parameters.

 

[adichy]

little confused, what does u and v represent :|, not sure what I am meant to be crossing

for c all i did was subtract the origin from a general point on the plane (x, y, 1-x-y)...is that wrong?

 

[LCKurtz]

little confused, what does u and v represent :|, not sure what I am meant to be crossing

for c all i did was subtract the origin from a general point on the plane (x, y, 1-x-y)...is that wrong?

No, not wrong. That is your R(x,y), using x and y as your parameters. Calculate R_x X R_y.

 

[adichy]

so i find δR/δx and δR/δY put them in a matrix then find the determinant ...

i got (i + j + k )dxdy

 

[LCKurtz]

so i find δR/δx and δR/δY put them in a matrix then find the determinant ...

i got (i + j + k )dxdy

Rx X R_y is <1 ,1 ,1> as you have calculated. In calculating surface area you would calculate

\iint_A |\vec R_x \times \vec R_y|\, dxdy

where A is the area domain in terms of your x-y integration.

 

[adichy]

um srry this mite b a really stupid question but how do u go about intergrating i + j + k or <1 ,1 ,1>, wudnt i hafta dot it with some vector :|

edit: jus realized there's a modulus >_<

i got the final answer to be \sqrt{3}/2

 

[LCKurtz]

Rx X R_y is <1 ,1 ,1> as you have calculated. In calculating surface area you would calculate

\iint_A |\vec R_x \times \vec R_y|\, dxdy

where A is the area domain in terms of your x-y integration.

um srry this mite b a really stupid question but how do u go about intergrating i + j + k or <1 ,1 ,1>, wudnt i hafta dot it with some vector :|

Please note that forum rules prohibit "baby-talk" text like um srry mite u wudnt hafta.

Notice the absolute value signs in the above formula. You need the length of that vector, which is a scalar.

The other common type of integral that use the "area vector" is a flux integral, where you are given a vector field and which in your example would have a formula like:

\iint_S \vec F\cdot d\vec R = \iint_S \vec F\cdot \vec R_x \times \vec R_y\, dxdy

In either case the integrand is a scalar.

 

[adichy]

noted: no more baby talk...can slip out sometimes unawares

Thanks for the help, much appreciated