Mathematical Physics | MCQs
گاؤس ڈائیورجنس تھیوریم ______ کو تبدیل کرتا ہے؟
A. Surface to line integral
B. None of these
C. Line to volume integral
D. Surface to volume integral
Explanation
The Gauss divergence theorem , also known as the divergence theorem, converts:
Volume integral to Surface integral
It states that the volume integral of the divergence of a vector field over a region is equal to the surface integral of the vector field over the boundary of that region.
Mathematically, it is expressed as:
∫∫∫_V (∇⋅F) dV = ∫∫_S F⋅dS
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A. Composite functions
B. Linear function
C. Multilinear function
D. None of these
Explanation
A tensor is a multilinear function. It maps vectors and their duals (covectors) to scalars , satisfying linearity in each argument. Tensors generalize the concept of scalars, vectors, and matrices in various dimensions. This making them essential in fields like physics and engineering. Show/Hide Explanation
A. √2
B. √3
C. 2√3
D. None of these
Explanation
Calculation: First, compute -2u + 2v:
− 2 u = ( − 4 , 4 , − 6 ) − 2 u = ( − 4 , 4 , − 6 )
2 v = ( 2 , − 6 , 8 ) 2 v = ( 2 , − 6 , 8 )
− 2 u + 2 v = ( − 4 + 2 , 4 − 6 , − 6 + 8 ) = ( − 2 , − 2 , 2 ) − 2 u + 2 v = ( − 4 + 2 , 4 − 6 , − 6 + 8 ) = ( − 2 , − 2 , 2 )
Norm Calculation: The norm (magnitude) of − 2 u + 2 v − 2 u + 2 v
∣ ∣ − 2 u + 2 v ∣ ∣ = ( − 2 ) 2 + ( − 2 ) 2 + 2 2 = 4 + 4 + 4 = 12 = 2 3 ∣∣ − 2 u + 2 v ∣∣ = ( − 2 ) 2 + ( − 2 ) 2 + 2 2 = 4 + 4 + 4 = 12 = 2 3 Answer: 2 3 2 3
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