Begin2.DVI

while the electric force acting on Qis Fe=QE (coul ) ( N coul ) (9 .146) The total force acting on the moving charge Qis F=F ...

hand in the direction around C so that the thumb points in the direction of the normal. Faraday’s law, obtained experimentally , ...

~r′=x′ˆe 1 +y′ˆe 2 +z′ˆe 3 to the point(x′, y′, z′)inside the volumeV. The vector~r−~r′ then points from the point(x′, y′, z′)to ...

In a similar fashion one can verify that ∂f ∂y= −3(y−y′) |~r−~r′|^5 and ∂f ∂z= −3(z−z′) |~r−~r′|^5 Using the above results verif ...

Example 9-14. The last Maxwell equation is hard to derive. Historically, Ampere^14 showed that for straight line currents the cu ...

and take the curl of the second and fourth of the Maxwell’s equations to obtain ∇× (∇× E
) = ∇(∇·E
)−∇^2 E
=∇× ( −∂ B
∂t ) =−∂t ...

9-5. Consider the vector field E
=r^12 eˆrin polar coordinates. (a) Show this vector field is irrotational. (b) Find a potenti ...

9-12. At all points (x, y )between the circles x^2 +y^2 = 1 and x^2 +y^2 = 9, the vector function F
= −yˆe 1 +xeˆ 2 x^2 +y^2 i ...

9-17. (Research project on orbital motion) Assume a mass m located at a position
r =reˆr experiences a central force F
=mf (r) ...

9-20. A right circular cone intersects a sphere of radius r as illustrated. Find the solid angle subtended by this cone. Right ...

9-27. In thermodynamics the internal energy U of a gas is a function of pressure P and volume V denoted by U =U(P, V ).If a gas ...

9-32. The problems below are concerned with obtaining a solution of Laplace’s equation for temperature T. Chose an appropriate ...

9-36. Express the Maxwell equations (9.161) as a system of partial differential equations.
9-37. Assume solutions to the Maxwe ...

9-42. Couloumb’s law states that the force between two charges q 1 and q 2 acts along the line joining the two charges and the ...

Chapter 10 Matrix and Difference Calculus The matrix calculus is used in the study of linear systems and systems of differ- enti ...

Properties of Matrices 1. Two matrices A= (aij)m×nand B= (bij)m×nhaving the same dimension are equal if aij =bij for all values ...

Example 10-1. Find the derivative and integral of the matrix A= [ 1 x sin x e−^2 x ] Solution Taking the derivative of each elem ...

equal the row dimension of B, then the matrices Aand Bcannot be multiplied. The matrix product of two matrices Aand B, having th ...

Example 10-2. If A= ( 1 2 3 4 5 6 ) 2 × 3 and B=   7 8 9 10 11 12 13 14 15 16 17 18   3 × 4 the ma- trices Aand Bsatisfy the ...

Example 10-3. Note that only in special cases is matrix multiplication com- mutative. One can say in general AB
=BA. Consider t ...