Consider a vector \(\bar{v}=\frac{\bar{r}}{r^3}\). The surface integral of this vector over the surface of a cube of side \(a\) and centered at the origin is :
zero
The surface integral of the vector itself is the vector integral \(\oint_S \bar{v}\,dS\) taken over the closed surface of the cube, with \(\bar{v}=\frac{\bar{r}}{r^3}\).
The field is odd under inversion through the origin: \(\bar{v}(-\bar{r})=\frac{-\bar{r}}{r^3}=-\bar{v}(\bar{r})\).
A cube centred at the origin is symmetric under \(\bar{r}\rightarrow-\bar{r}\): every area element \(dS\) at \(\bar{r}\) on one face has an equal element at \(-\bar{r}\) on the opposite face.
The contributions of each such pair are \(\bar{v}(\bar{r})\,dS+\bar{v}(-\bar{r})\,dS=0\), so they cancel pairwise over the whole surface: \(\oint_S \bar{v}\,dS=0\).
The same result follows for the other vector surface integral \(\oint_S d\bar{S}\times\bar{v}=\int_V(\vec{\nabla}\times\bar{v})\,dV\), because \(\bar{v}=-\vec{\nabla}\left(\frac{1}{r}\right)\) is a gradient and hence curl-free: \(\vec{\nabla}\times\bar{v}=0\).
Hence, the answer is zero.
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
