GATE (TF) Textile 2016 Question Paper Solution | GATE/2016/TF/27

Question 27 (Textile Engineering & Fibre Science)

Let $f(x,y,z) = \frac{1}{\sqrt{(x^2+y^2+z^2)}}$ . The value of $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}$ is equal to __-0.01:0.01___.

[Show Answer]

Write Here

Given-
$f(x,y,z) = \frac{1}{\sqrt{(x^2+y^2+z^2)}}$
$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}$ =?
$\frac{\partial^2 f}{\partial x^2}$
$\frac{\partial f}{\partial x}=\frac{-1}{{2(x^2+y^2+z^2)}^\frac{3}{2}}\times (2x)$
$\frac{\partial f}{\partial x}=\frac{-1}{{(x^2+y^2+z^2)}^\frac{3}{2}}\times (x)$
$\frac{\partial^2 f}{\partial x^2}=\frac{-1}{{(x^2+y^2+z^2)}^\frac{3}{2}}\times 1+2x \times \frac{3}{2{(x^2+y^2+z^2)}^\frac{5}{2}}$
Similarily-
$\frac{\partial^2 f}{\partial y^2}=\frac{-1}{{(x^2+y^2+z^2)}^\frac{3}{2}}\times 1+2y \times \frac{3}{2{(x^2+y^2+z^2)}^\frac{5}{2}}$
and
$\frac{\partial^2 f}{\partial z^2}=\frac{-1}{{(x^2+y^2+z^2)}^\frac{3}{2}}\times 1+2z \times \frac{3}{2{(x^2+y^2+z^2)}^\frac{5}{2}}$
By adding-
$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}$ =0 (ANS)

Previous Question

Next Question

Frequently Asked Questions | FAQs

GATE Textile Engineering and Fibre Science (TF) Question Papers | GATE Textile Question Answer | GATE Textile Solved Question Papers | GATE Textile Papers | GATE Textile Answer Key