Deriving the surface area equation

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SUMMARY

The discussion focuses on deriving the surface area equation by analyzing the area formed by tangent vectors "a" and "b" on a curved surface. The user initially attempts to treat this area as a rectangle but realizes that the vectors are not perpendicular due to the curvature of the surface. This leads to the conclusion that the area must be calculated by projecting the tangent vectors onto the xy-plane, where they may not maintain their perpendicularity. The extra term encountered in the calculations suggests a need for further investigation into its behavior, particularly why it should approach zero.

PREREQUISITES
  • Understanding of vector calculus, specifically tangent vectors on curved surfaces.
  • Familiarity with surface area calculations in multivariable calculus.
  • Knowledge of projection techniques in coordinate systems.
  • Basic grasp of differential geometry concepts.
NEXT STEPS
  • Study the derivation of surface area for parametric surfaces in multivariable calculus.
  • Learn about the projection of vectors onto different coordinate planes.
  • Investigate the implications of curvature on vector relationships in differential geometry.
  • Explore the concept of differential elements in calculus and their applications in surface area calculations.
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus and differential geometry, as well as educators teaching surface area derivations and vector calculus concepts.

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consider the following image

4IrsvlR.png


(the red is the surface area element and the green is the differential element that I'm integrating over)

when we derived this in class, we treated the area formed by vectors a and b, as the area of a parallelogram. the thing is, a and b should be at right angles implying that rest of the angles should be right angles as well. if that's the case, then why can't i just treat the area formed from a and b, as a rectangle?

well, that's what i attempted to try but failed. i get all the appropriate terms as they do in the surface area formula, except i get an extra term as shown in the link below, implying, perhaps.. that that extra term must go to zero. but i don't know why it should.

if I'm incorrect in treating the shape formed from a and b as a rectangle, why is this incorrect?

calculations:

http://i.imgur.com/6MFO4xI.jpg
(image was too large to put here)
 
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It's very difficult to read that- the colors do not show up on black very well! But to answer your question, the "a" and "b" on the surface are NOT vector IN the surface because it is not flat- they are tangent vectors to a curved surface and so do NOT form a rectangle in the surface. In order to calculate and area we have to project down to the xy- plane and the vectors are no longer necessarily perpendicular there.
 

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