Normal to parameterised surface

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Homework Help Overview

The discussion revolves around finding a normal vector to a parameterized surface defined by the equations \( (u,v) \) leading to a surface in three-dimensional space. The problem also involves writing a double integral for the surface area and determining the equation of the tangent plane at a specific point.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore methods for finding a normal vector using the cross product of tangent vectors. There is discussion on how to compute the Jacobian and its relation to the double integral for surface area. Questions arise about the appropriate techniques for deriving tangent vectors and the implications of using Jacobians versus other methods.

Discussion Status

Some participants have begun to clarify their understanding of the problem, with one noting the first part of the problem is resolved. Others are actively seeking guidance on computing the Jacobian and confirming the values of \( u \) and \( v \) for the tangent plane equation.

Contextual Notes

There is mention of constraints regarding the methods learned in class, with some participants expressing confusion over the application of certain concepts such as the Jacobian and the fundamental vector product.

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Homework Statement


surface : $(u,v) is (e^u,(v^2)*(e^(2u)),2e^(-u) +v)
0<=u<=3 -4<=v<=4
find a normal in terms of u and v
write down a double integral for the surface(dont solve)
find equation of tangent plane to surface at (1,4,0)

Homework Equations





The Attempt at a Solution


I have no idea where to start

any help please?
 
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Given two vectors, what is one easy way to get a vector normal to both?
 
cross product
 
Can you find two non-collinear vectors tangent to the surface at a single point? Since both vectors are tangent, their cross-product should point normal to the surface.
 
ok got the first part thanks, now for the area, i know in the double integral has |J| in it somehow, how wuld i find it. I know from a mate u can do |Tu X Tv|, but i don't think we learned this, and i was wondering how wuld u do this by using jacobians, if possible.

im confused coz we abs(del(x,y,z)/del(u,v)| and i don't know how to solve this
 
One very good way to do all three of those questions is to find the "fundamental vector product". To do that, you find the vector derivatives of "$(u,v)" with respect to u and v and take the cross product of those two vectors.

The derivative vectors are, of course, tangent vectors to the "coordinate curves" v= constant and v= constant at each point and so lie in the tangent plane at each point. Therefore, their cross product is normal to the tangent plane and so to the surface at that point. Further, the length of that vector is the Jacobian (it is the |Tu x Tv| you mention). And, of course, if you know a normal at a point, it is easy to write down the equation of the tangent plane at that point.

By the way, have you determined u and v for the point (1, 4, 0)?
 
the points are u=0 and v=-2, so is the tangent plane is Tu(0,2)*(u-0)+Tv(0,2)*(v-2)=0
 

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