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Vector calculus (Read 3119 times)
Gerrit-Jan Linker
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Vector calculus
09.08.11 at 20:51:27
 
Vector calculus
 
Vector calculus is the branch of mathematics that is concerned with integration and differentiation of vector functions.
 
Differential operations:
 
Gradient:
Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
grad(f) =▼f  
The gradient works on a scalar field and gives a vecxtor field.
 
Curl:
Measures the tendency to rotate about a point in a vector field.
curl(f)=▼xf
 
Divergence:
Maps vector fields to scalar fields.
div(f)=▼.f
 
 
Laplacian:
∆f=▼2f=▼.▼f  
 
Definitions:
▼ ≡ (∂/∂x, ∂/∂y , ∂/∂z) ≡ i∂/∂x + j∂/∂y + k∂/∂z
2 ≡ ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
 
Source:
http://en.wikipedia.org/wiki/Vector_calculus
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« Last Edit: 09.08.11 at 21:32:37 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Gerrit-Jan Linker
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Posts: 75
Gradient examples
Reply #1 - 09.08.11 at 22:37:47
 
Gradient examples
 
grad (1/x) = ▼ x-1 = (∂/∂x, ∂/∂y , ∂/∂z) x-1 = (-x-2,0,0)
 
grad (1/x+y) = ▼ (x+y)-1 = (∂/∂x, ∂/∂y , ∂/∂z) (x+y)-1 = ( -(x+y)-2 , -(x+y)-2 , 0)
 
grad (1/(x2+y2) = ▼ (x2+y2) = (∂/∂x, ∂/∂y , ∂/∂z) (x2+y2) = ( -2x/(x2+y2) , -2y/(x2+y2) , 0)
 
grad (1/√(x2+y2) = ▼ (x2+y2)-0.5 = (∂/∂x, ∂/∂y , ∂/∂z) (x2+y2)-0.5 = ( -x(x2+y2)-1.5 , -y(x2+y2)-1.5 , 0)
 
grad (1/√(x2+y2+z2) = ▼ (x2+y2+z2)-0.5 = (∂/∂x, ∂/∂y , ∂/∂z) (x2+y2+z2)-0.5 = ( -x(x2+y2+z2)-1.5 , -y(x2+y2+z2)-1.5 , -z(x2+y2+z2)-1.5)
 
See also:
http://www.wolframalpha.com/input/?i=grad+1%2F%28x^2%2By^2%2Bz^2%29^0.5
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« Last Edit: 10.08.11 at 15:08:19 by Gerrit-Jan Linker »  

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