Find if a Vecotr field is perpendicular to a curve

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Homework Help Overview

The discussion revolves around determining if a vector field is perpendicular to a curve, as presented in a specific homework problem. The subject area includes vector calculus and differential geometry.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the concept of perpendicularity between a vector field and a curve, questioning whether this applies at every point on the curve. They discuss the use of the dot product to assess perpendicularity and consider the implications of working with an implicitly defined curve.

Discussion Status

The discussion is active, with participants providing insights on how to determine perpendicularity through mathematical concepts such as the dot product and the tangent vector. There are multiple approaches being considered, including the use of the implicit function theorem and cross products.

Contextual Notes

Participants note that the curve is given implicitly, which may complicate the determination of the tangent vector. There is an acknowledgment of the need for further clarification on the problem setup and definitions involved.

swraman
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Homework Statement



http://math.berkeley.edu/~teleman/53f08/review2.pdf
Question #4

Homework Equations



Not sure

The Attempt at a Solution



I have no idea...
I really don't care about the answer, I have that, but I just don't know how to find out if a vector field is perpendicular to a curve.
Thanks
 
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Do you mean perpendicular at every point on the curve? You determine whether a single vector is perpendicular to a curve at a given point by taking the dot product of the vector with the tangent vector to the curve at that point. Can you generalize that to a vector field?
 
If you want to find out if a vector field is perpendicular you have to check that the scalar product of the tangent vector of the curve and the Vector Field vanishes at every point of the curve.

Since you have given the curve only implicitly you would probably need implicit function theorem to actually get the tangent vector.
 
The easy way to get a tangent direction is to take the cross product of the normals of the surfaces defined by the two equations.
 

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