Gauss-divergence theorem for volume integral of a gradient fieldI 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

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Gauss-divergence theorem for volume integral of a gradient fieldI 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

Sheldon Castillo · Accepted Answer

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 · Accepted Answer

Step 1The 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 2The 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      →

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

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