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[quote="Anonymous"]1. (The parametrization of a curve and its normal and curvature) Let C be a curve
parametrized by the vector-valued function <math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mo>:</mo><mrow><mo>[</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>&rarr;</mo><msup><mrow><mi>&Ropf;</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>

<math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo>,</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo>)</mo><mtext>&ThinSpace;</mtext><mo>,</mo><mo>-</mo><mn>1</mn><mo>&le;</mo><mi>t</mi><mo>&le;</mo><mn>1.</mn><mtext>&NewLine;</mtext></mrow></math>

(a) Find the arc-length parametrization of the curve.
(b) Find the principal unit normal vector of C at<math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mn>.</mn></mrow></math>
(c) Find the curvature of C at<math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mn>.</mn></mrow></math>[/quote]

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[quote="Anonymous"]1. (The parametrization of a curve and its normal and curvature) Let C be a curve parametrized by the vector-valued function <math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mo>:</mo><mrow><mo>[</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>→</mo><msup><mrow><mi>&Ropf;</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> <math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo>,</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo>)</mo><mtext> </mtext><mo>,</mo><mo>-</mo><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>1.</mn><mtext>&NewLine;</mtext></mrow></math> (a) Find the arc-length parametrization of the curve. (b) Find the principal unit normal vector of C at<math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mn>.</mn></mrow></math> (c) Find the curvature of C at<math><mrow><mover><mrow><mi>r</mi></mrow><mo>&rharu;</mo></mover><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mn>.</mn></mrow></math>[/quote]

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[大學] 有關向量值函數的問題

由訪客於星期三五月 05, 2010 1:29 am

1. (The parametrization of a curve and its normal and curvature) Let C be a curve
parametrized by the vector-valued function

(a) Find the arc-length parametrization of the curve.
(b) Find the principal unit normal vector of C at
(c) Find the curvature of C at

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[大學] 有關向量值函數的問題