Begin2.DVI
The ∇operator and integration 16. ∫∫∫ V ∇f dV = ∫∫ S feˆndS Special case of divergence theorem 17. ∫∫∫ V ∇× A dV = ∫∫ S ˆen×A d ...
of a general point (x, y, z)can be expressed in terms of the curvilinear coordinates (u, v, w )by utilizing the transformation e ...
partial derivative ∂u∂r .Similarly, the curves r =r (c 1 , v, c 3 )and r =r (c 1 , c 2 , w )are coor- dinate curves and hav ...
The vector grad u(x, y, z )is a vector normal to the surface u=c 1 .A unit normal to the u=c 1 surface has the form Eu= grad u ...
In rectangular coordinates consider the family of surfaces x=c 1 , y =c 2 , z =c 3 , where c 1 , c 2 , c 3 take on the integer v ...
represents a straight line parallel to the z-axis which is normal to the xy plane at the point r=c 1 , θ =c 2 .The curve r =r ...
Observe that this is an orthogonal system where gij = 0 for i=j. The surface r=c 1 is a cylinder, whereas the surface θ =c 2 is ...
where c 1 , c 2 , c 3 are constants. These curves are, respectively, circles of radius c 1 sin c 2 , meridian lines on the surfa ...
The transformation equations (8.68) are obtained from the geometry in figure 8-19. These equations are x=rcos θ y=rsin θ+ηcos α ...
Spherical coordinates (ρ, θ, φ ) : x=ρsin θcos φ, ρ ≥ 0 y=ρsin θsin φ, 0 ≤φ≤ 2 π z=ρcos θ, 0 ≤θ≤π ds^2 =h^2 ρdρ^2 +h^2 θdθ^2 +h^ ...
Elliptic cylindrical coordinates (ξ, η, z ) : x= cosh ξcos η, ξ ≥ 0 y= sinh ξsin η, 0 ≤η≤ 2 π z=z, −∞ < z < ∞ ds^2 =h^2 ξd ...
Here the quantities A 1 , A 2 , A 3 represent the components of the vector field Ain rect- angular coordinates, and Ar, A θ, A ...
Example 8-18. Express the vector A= 2yˆe 1 +zˆe 2 + 2xˆe 3 in cylindrical coordinates. Solution The rectangular components of A ...
where Au, A v, A w denote the components of A in the new coordinate system and are functions of these coordinates. The transfor ...
Since ∂φ ∂x eˆ 1 +∂φ ∂y ˆe 2 +∂φ ∂z eˆ 3 = ( ∂φ ∂u ∂u ∂x +∂φ ∂v ∂v ∂x +∂φ ∂w ∂w ∂x ) ˆe 1 + ( ∂φ ∂u ∂u ∂y + ∂φ ∂v ∂v ∂y + ∂φ ∂w ...
The first term in equation (8.94) can be expanded, and ∇(Fuˆeu) = ∇(Fu)·ˆeu+Fu∇(ˆeu) = 1 h 1 ∂F u ∂u +Fu∇[h^2 h^3 (∇v)×(∇w)] (Se ...
Since curl grad u= 0,the above simplifies to ∇× (Fuˆeu) = ∇(Fuh 1 )× ˆeu h 1 = [ 1 h 1 ∂(Fuh 1 ) ∂u ˆeu+^1 h 2 ∂(Fuh 1 ) ∂v ˆev+ ...
Example 8-20. The Laplacian in rectangular coordinates (x, y, z)is given by ∇^2 U= ∂^2 U ∂x^2 + ∂^2 U ∂y^2 + ∂^2 U ∂z^2 (8 .101) ...
Exercises 8-1. Sketch some level curves φ=kfor the given values of kand then find the gradient vector. (i) φ= 4x− 2 y, k =− 2 , ...
8-7. Find the minimum distance between the lines defined by the parametric equations L 1 : x=τ− 1 , y =−τ+ 16, z = 2τ− 2 L 2 : ...
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