The value of integral ∯_{S} \overrightarrow{r} \cdot \overrightarrow{n} d S over the closed surface S bounding a volume, where r=x \hat{i}+y \hat{i}+z \hat{k} is the position vector and \overrightarrow{n} is the normal to the surface {S}, is

GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 43 of 65

Marks: +1, -0

The value of integral

∯_{S} {r}↖{→} ⋅ {n}↖{→} d S

over the closed surface

S

bounding a volume, where

r=x {i}↖{∧}+y {i}↖{∧}+z {k}↖{∧}

is the position vector and

{n}↖{→}

is the normal to the surface

{S}

, is

Solution:

By Gauss Divergence Theorem

∯↙{S} {r}↖{→} ⋅ {n}↖{→} d S=∫∫∫_{V}(\ov{V} ⋅ {r}↖{→}) d V

(\ov{V} ⋅ {r}↖{→})=3

Hence,

∯_{S} {r}↖{→} ⋅ {n}↖{→} d S=3∫∫∫_{V} d V=3 {V}

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