Begin2.DVI
around an arbitrary closed path is zero. Note that Stokes’ theorem, with ∇× F= 0 , implies that the line integral around an ar ...
The line integral (9.45) can be expressed as the sum of the line integrals along the straight line paths P 0 P 1 , P 1 P 2 , P 2 ...
∂φ ∂y =F^2 (x, y, z^0 ) + ∫z z 0 ∂F 2 (x, y, z ) ∂z dz =F 2 (x, y, z 0 ) + F 2 (x, y, z ) z z 0 =F 2 (x, y, z 0 ) + F 2 (x, y, z ...
where f 1 , f 2 , f 3 are arbitrary functions which have been held constant during the partial differentiation process. Now add ...
ψis any scalar function, is also a vector satisfying F = curl V∗.This result is verified by using the distributive property of ...
and thereby simplify the integral (9.55) to the form ∂V 2 ∂x − ∂V 1 ∂y = ∫z z 0 ∂F 3 ∂z dz + ∂f 2 ∂x − ∂f 1 ∂y =F 3 (x, y, z)−F ...
This force is derivable from the potential function φ= k ρ, where ρ= √ (x−x 1 )^2 + (y−y 1 )^2 + (z−z 1 )^2 is the distance betw ...
Example 9-3. Multiply both sides of Newton’s second law F =ma =md (^2) r dt^2 by dr dt and then integrate from P 0 (x 0 , y ...
By Stokes’ theorem ∫∫ S 1 F·dS 1 = ∫∫ S 1 (∇× V)·dS 1 = ∫ C ©V ·dr and ∫∫ S 2 F·dS 2 = ∫∫ S 2 (∇× V)·dS 2 = ∫ C ©− V· ...
Definition: (Equipotential curves) If F =F(x,y )is a given conservative vector field with potential φ(x,y ),then the family of ...
of the family. Let ψ(x, y ) = c∗ denote the family of orthogonal trajectories to the family of equipotential curves φ(x, y ) = c ...
This differential equation is derived by requiring the direction of the vector field at an arbitrary point (x, y )have the same ...
Vector Fields Irrotational and Solenoidal If in addition to being conservative, the two-dimensional vector field given by F=M(x ...
That is grad φ= ∂φ ∂x ˆe 1 +∂φ ∂y ˆe 2 =∂ψ ∂y ˆe 1 −∂ψ ∂x ˆe 2. (9 .70) Differentiate the Cauchy–Riemann equations (9.69) and sh ...
In various branches of science and engineering, the quantities φand ψhave many different physical interpretations. For example, ...
In a cylindrical (r, θ, z )coordinate system , the Laplace equation takes the form ∇^2 U=∂ (^2) U ∂r^2 +^1 r ∂U ∂r +^1 r^2 ∂^2 U ...
satisfied by the equipotential surfaces is obtained by differentiating φ(x, y, z) = cto obtain the exact differential dφ = ∂φ ∂x ...
c 2 are the two parameters. Two methods for obtaining independent integrals of equations (9.83) are now presented. Theory of Pro ...
This is an equation where the variables can be separated and then an integration produces another independent family of surfaces ...
Example 9-7. The field lines of the vector field F=F(x, y, z ) = yˆe 1 +xˆe 2 +zˆe 3 are determined from the differential syst ...
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