Begin2.DVI

and an integration of each of these functions produces μ 3 =^1 2 xz +f(y, z ) μ 3 =− 1 2 yz +g(x, z) μ 3 =^12 (x−y)z+h(x, y ), w ...

A general cone is described by a line having one point fixed in space which is free to rotate. The figure 9-5 illustrates two co ...

( v) A unit sphere about the origin. (vi) Connect the points on the boundary of dS to the origin and form a cone which will inte ...

Consider the special surface integral ∫∫ S dω = ∫∫ S dΩ r^2 = ∫∫ S ˆen·
r r^3 dS = ∫∫ S
r ·dS
r^3 where the surface Sencloses ...

This produces an element of area dS = (rdθ)(2πr sin θ) = 2 πr^2 sin θ dθ for 0 ≤θ≤θ 0 The total surface area of the spherical ca ...

∆M. It is assumed that the velocity is the same at all points over the tiny element of surface area. In a time interval ∆t, the ...

If dρdt = 0,then the fluid is an incompressible fluid, and the velocity field is solenoidal. If the fluid flow is also irrotatio ...

which by the divergence theorem can be expressed in the form ∫∫ S
q ·dS
= ∫∫∫ V div
q dV. (9 .97) Substituting the heat flow g ...

other mass takes place in a plane. Construct a set of x, y axes with origin located at the center of mass of M. Further, let eˆr ...

With reference to figure 9-8, a conic section is defined in terms of the ratio OPP D = , where OP =r, and P D = 2q−rcos θ. From ...

Note that the Newton law of gravitation implies that the derivative given by equation (9.107) is zero. That is, if md (^2)
r dt ...

where p=h^2 /GM and =C/GM. This result is known as Kepler’s first law and implies that all the planets of the solar system descr ...

where Ais the area of the ellipse and τ is the period of one orbit. The area of an ellipse is given by the formula A=πab, where ...

There are three cases to consider. Case 1 The roots of characteristic equation (9.136) are real and unique. If r 1 , r 2 are the ...

where αand β are scalar constants is obtained by first solving the homogeneous equation d^2
y dt^2 +α d
y dt +β
y=
0 given by ...

Case 2 If α^2 −β^2 = 0, the characteristic equation has the repeated roots λ=−α. The first root gives the first member of the fu ...

Maxwell’s Equations James Clerk Maxwell (1831-1874), a Scottish mathematician, studied properties of electric and magnetic field ...

Note 2: The product μ 0 0 =^1 c^2 , where c= 3 ×(10)^10 cm/sec is the speed of light. It will be demonstrated later in this chap ...

Example 9-10. Consider the special case of a single point charge qlocated at the origin. The electric field due to this point ch ...

If the discrete number of n-charges q 1 ,...,qnwere replaced by a continuous dis- tribution of charges inside the surface, then ...