Find the volume integral of the gradient field. The author directly uses the Gauss-divergence theorem to relate the volume integral of gradient of a scalar to the surface integral of the flux through the surface surrounding this volume, i.e. int_{CV} grad phi dV=int_{delta CV} phi dS

Ciara Rose

Ciara Rose

Answered question

2022-07-14

Gauss-divergence theorem for volume integral of a gradient field
I need to make sure that the derivation in the book I am using is mathematically correct. The problem is about finding the volume integral of the gradient field. The author directly uses the Gauss-divergence theorem to relate the volume integral of gradient of a scalar to the surface integral of the flux through the surface surrounding this volume, i.e.
C V ϕ d V = δ C V ϕ d S

Answer & Explanation

Sheldon Castillo

Sheldon Castillo

Beginner2022-07-15Added 10 answers

Step 1
C V ϕ d V = i x ^ i C V ϕ x i d V = i x ^ i C V ( ϕ x ^ i ) d V = i x ^ i C V ϕ x ^ i d S = i x ^ i C V ϕ ( d S ) i = C V ϕ i ( d S ) i x ^ i =   C V ϕ d S  
One interesting application of this identity is the Archimedes Principle derivation ( the force magnitude over a body in a fluid is equal to the weight of the mass of fluid displaced by the body ):
{ P a t m . : Atmospheric Pressure. ρ : Fluid Density. g : Gravity Acceleration   9.8   m s e c 2 . z : Depth. m f l u i d . : ρ V b o d y = ρ C V d V
C V ( P a t m . + ρ g z ) ( d S ) = C V ( P a t m . + ρ g z ) d V = C V ρ g z ^ d V = m f l u i d g z ^
Raynor2i

Raynor2i

Beginner2022-07-16Added 6 answers

Step 1
The statement is true. It is typically proved using following properties of vectors.
Two vectors p , q R n equals to each other if and only if for all vectors r R n , r p = r q .
Back to our original identity. For any constant vector k , we have
k ( C V ϕ d V ) = C V ( ϕ k ) d V = div. theorem C V ϕ k d S = k ( C V ϕ d S )
Step 2
The first equality holds because k ϕ = ( ϕ k ) ϕ ( k ). Additionally, since k is a constant vector, k = 0. Hence, k ϕ = ( ϕ k ).
Since this is true for all constant vector k, the two vectors defined by the integrals equal to each other. i.e.
C V ϕ d V = C V ϕ d S

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