Surface Integral

Calculus Level 4

F ⃗ = x ( y − z ) ı ^ + y ( z − x ) ȷ ^ + z ( x − y ) k ^ \Large{\vec{F} = x(y-z)\hat{\imath} + y(z-x)\hat{\jmath} + z(x-y)\hat{k}}

Determine the surface integral of the vector field F ⃗ \vec{F} over a unit sphere centered on the origin.

Note: ı ^ , ȷ ^ , k ^ \hat{\imath}, \hat{\jmath},\hat{k} are unit vectors associated with the three Cartesian coordinate axes ( x , y x,y , and z z respectively).


The answer is 0.

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
Sep 28, 2016

According to the divergence theorem See here , the surface integral of a vector field over a closed surface is equal to the integral of the divergence of the vector field over the volume enclosed by the surface. The divergence can be expressed as:

▽ ⋅ F ⃗ = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z = ( y − z ) + ( z − x ) + ( x − y ) = 0 \bigtriangledown \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = (y - z) + (z - x) + (x- y) = 0

Since the divergence of the vector field is zero, the surface integral is zero. Incidentally, this would be true for any closed surface.

4 Helpful 1 Interesting 1 Brilliant 0 Confused

@Steven Chase Sir can you provide me a code for this problem
Thanks in advance
Hope I am not disturbing you.

NJ STAR - 10 months, 3 weeks ago

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I take the location of the person as x = 0 x = 0 . The ball lands 14.24 14.24 meters to the left of the person. 

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import math

g = 9.8

dt = 10.0**(-6.0)
t = 0.0

x = 0.0
y = 2.0

xd = 10.0
yd = 10.0

xdd = 0.0
ydd = -g

###################################

# Going to wall

while x <= 4.0:

    x = x + xd*dt
    y = y + yd*dt

    xd = xd + xdd*dt
    yd = yd + ydd*dt

    t = t + dt

###################################

# Bouncing off of wall

xd = -xd

###################################

# Final part of trajectory

while y >= 0.0:

    x = x + xd*dt
    y = y + yd*dt

    xd = xd + xdd*dt
    yd = yd + ydd*dt

    t = t + dt

###################################

print dt
print x
print (x/4.0)

#>>> 
#1e-05
#-14.2433
#-3.560825
#>>> ================================ RESTART ================================
#>>> 
#1e-06
#-14.2431799996
#-3.56079499991
#>>>

Steven Chase - 10 months, 3 weeks ago

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