\(\vec r\) की दिशा में 1 / r का दिशात्मक अवकलज क्या है?

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Official Sr. Teacher Gr II NON-TSP MATHEMATICS (Held on :29 Oct 2018)
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  1. \(\frac {1}{r^2}\)
  2. \(-\frac {1}{r^2}\)
  3. \(\frac {1}{r^3}\)
  4. \(-\frac {\vec r}{r^3}\)

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Option 2 : \(-\frac {1}{r^2}\)
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अवधारणा :

माना कि f(r) एक फलन है तो फलन f(r) के दिशात्मक अवकलज को निम्न द्वारा दिया जाता है: \(\nabla f(r) = f'(r) \frac{\vec r}{|\vec r|}\)

गणना :

माना कि f(r) = 1/r

जैसा कि हम जानते हैं कि, यदि f(r) एक फलन है तो फलन f(r) के दिशात्मक अवकलज को निम्न द्वारा दिया जाता है: \(\nabla f(r) = f'(r) \frac{\vec r}{|\vec r|}\)

⇒ f'(r) = \(- \frac{1}{r^2}\)

यहाँ, \(r = |\vec r|\)

\(\nabla f(r) = - \frac{1}{r^2} \cdot \frac{\vec r}{r} = - \frac{\vec r}{r^3}\)

यहाँ, हमें \(\vec r\) की दिशा में f(r) के दिशात्मक अवकलज का पता लगाना होगा

\(\nabla f(r)\dot\ \frac{\vec r}{r}\ = - \frac{\vec r}{r^3}\ \dot\ \frac{\vec r}{r}\ = -\frac{\ r^2}{r^4}\ = - \frac{1}{r^2}\)

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