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definition of a directional derivative. I may add this material which talks about a vector part and a point of application as comprising a tangent vector which is just these two pair of points in
Euclidean space. I don't like this definition however. I need a little more mathematical sophistication, but I intend to cover tangent vectors as
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Please provide suggestions for improvement! I think I may have made the first major revision to this article. I reorganized the old material into the intro and added a definition and properties as well as reference. I may add more later. Gray's definition of a tangent vector is actually prior to his
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I do not think that this material is too technical. A tangent is part of elementary calculus, but a tangent vector is part of vector calculus. Therefore, our readers will have some very small bit of mathematical sophistication (like
680:{\displaystyle \mathbf {T} (0)={\frac {\mathbf {r} ^{\prime }(0)}{|\mathbf {r} ^{\prime }(0)|}}=\left.{\frac {(2t,2e^{2t},\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.}
422:{\displaystyle \mathbf {T} (0)={\frac {\mathbf {r} ^{\prime }(0)}{|\mathbf {r} ^{\prime }(0)|}}=\left.{\frac {(2t,2e^{2t},\sin {t})}{\sqrt {4t^{2}+e^{2t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.}
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The example only makes sense if the denominator evaluates to the value 2 at t=0; e.g., (4t + 4e + sint ) = 2. Arkadi is correct.
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Here the bottom part is not equal to the vector length. AFAIK it must be
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elements of a tangent space a differentiable manifold.
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
