Mathematics-I (3110014)

Winter-2019

Question-5c

BE | Semester-1 Winter-2019 | 17-01-2020

Q5) (c)

If $u = f (x - y, y - z, z - x)$ then show that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ .

$Let, x - y = l$

$\frac{\partial l}{\partial x} = 1$

$\frac{\partial l}{\partial y} = - 1$

$Let, y - z = m$

$\frac{\partial m}{\partial y} = 1$

$\frac{\partial m}{\partial z} = - 1$

$Let, z - x = n$

$\frac{\partial n}{\partial x} = - 1$

$\frac{\partial n}{\partial z} = 1$

$We have, u = f (x - y, y - z, z - x) = f (l, m, n)$

$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial l} \frac{\partial l}{\partial x} + \frac{\partial u}{\partial n} \frac{\partial n}{\partial x}$

$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial l} (1) + \frac{\partial u}{\partial n} (- 1)$

$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial l} - \frac{\partial u}{\partial n}$

$\frac{\partial u}{\partial y} = \frac{\partial u}{\partial l} \frac{\partial l}{\partial y} + \frac{\partial u}{\partial m} \frac{\partial m}{\partial y}$

$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial l} (- 1) + \frac{\partial u}{\partial m} (1)$

$\frac{\partial u}{\partial z} = - \frac{\partial u}{\partial l} + \frac{\partial u}{\partial m}$

$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial m} \frac{\partial m}{\partial z} + \frac{\partial u}{\partial n} \frac{\partial n}{\partial z}$

$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial m} (- 1) + \frac{\partial u}{\partial n} (1)$

$\frac{\partial u}{\partial z} = - \frac{\partial u}{\partial m} + \frac{\partial u}{\partial n}$

$Now, \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z}$

$No = \frac{\partial u}{\partial l} - \frac{\partial u}{\partial n} - \frac{\partial u}{\partial l} + \frac{\partial u}{\partial m} - \frac{\partial u}{\partial m} + \frac{\partial u}{\partial n}$

$No = 0$

$Hence, proved .$

Mathematics-I (3110014)

Q5) (c)

If $u = f (x - y, y - z, z - x)$ then show that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ .

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Mathematics-I (3110014)

Q5) (c)

If u=fx-y, y-z, z-x then show that ∂u∂x+∂u∂y+∂u∂z=0.

Questions

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If $u = f (x - y, y - z, z - x)$ then show that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ .