Understanding the Unit Normal Vector in Multivariable Differential Calculus

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SUMMARY

The discussion focuses on the proof of the equation involving the unit normal vector N(t) in multivariable differential calculus, specifically in the context of space curves defined by the vector function r(t). The equation presented is r'(t) x r''(t) x r'(t) = N(t) / ||r'(t) x r''(t) x r'(t)||. Here, r'(t) represents the tangent vector, while r''(t) indicates the change in the tangent vector, which lies in the osculating plane. The geometric behavior of the cross product is crucial for understanding the direction of the normal vector.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically cross products.
  • Familiarity with space curves and their representations in R³.
  • Knowledge of tangent and normal vectors in differential geometry.
  • Basic proficiency in multivariable calculus, particularly derivatives of vector functions.
NEXT STEPS
  • Study the properties of cross products in vector calculus.
  • Learn about the Frenet-Serret formulas for curves in R³.
  • Explore the concept of curvature and torsion in differential geometry.
  • Investigate the geometric interpretation of normal vectors in space curves.
USEFUL FOR

Students and educators in multivariable calculus, mathematicians focusing on differential geometry, and anyone seeking to deepen their understanding of vector functions and their geometric interpretations.

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This question comes from my multivariable differential calculus course. I cannot figure how to prove that the following is true...

How does...

__r'(t) x r''(t) x r'(t)_ = N(t) ?
||r'(t) x r''(t) x r'(t)||

any help would be appreciated...thanks
 
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What are r and N?
 
issisoccer10 said:
This question comes from my multivariable differential calculus course. I cannot figure how to prove that the following is true...

How does...

__r'(t) x r''(t) x r'(t)_ = N(t) ?
||r'(t) x r''(t) x r'(t)||

any help would be appreciated...thanks

Just think about how the cross product behaves geometrically. I'm assuming r is a map from R into R3 defining a space curve and that N is a normal vector to the curve, defined by being normal to the tangent vector at each point. r'(t) would then define a tangent vector at the point r(t). r''(t) describes the change in the tangent vector at that point, so for an arbitrarily small window, it should lie in the osculating plane of the curve. What direction does the cross product of these two vectors go in with respect to these two vectors? What then happens when you take the cross product of this vector with the tangent vector?
 

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