By D.A. Dubin

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**Extra resources for Solvable models in algebraic statistical mechanics**

**Example text**

How these ﬁelds interact with each other and with materials is what electromagnetics is all about. 1 Scalar Fields A scalar ﬁeld is a ﬁeld of scalar variables; that is, if for any point in space, say (x,y,z), we know the function f (x,y,z), then f is the scalar ﬁeld. This may represent a temperature distribution, potential, pressure, or any other scalar function. For example, f ðx; y; zÞ ¼ x2 þ y2 þ 5z2 ð1:45Þ is a scalar ﬁeld. 19. It shows a topographical map in which contour lines show various elevations.

6) The scalar product is commutative and distributive: Á Á ðcommutativeÞ A Á ðB þ CÞ ¼ A Á B þ A Á C ðdistributiveÞ A B¼B A ð1:24Þ ð1:25Þ The scalar product can be written explicitly using two vectors A and B in Cartesian coordinates as À Á À Á ^x Bx þ ^y By þ ^z Bz x Ax þ ^ y Ay þ ^z Az A B¼ ^ Á Á Á Á Á Á Á Á Á Á Á ¼^ x ^ x Ax Bx þ ^x ^y Ax By þ ^x ^z Ax Bz þ ^y ^x Ay Bx þ ^y ^y Ay By þ ^y ^z Ay Bz þ ^z ^x Az Bx þ ^z ^y Az By þ ^z ^z Az Bz ð1:26Þ From properties (2) and (3) and since unit vectors are of magnitude 1, we have Á Á Á Á Á Á ^x ^y ¼ ^x ^z ¼ ^y ^z ¼ ^y ^x ¼ ^z ^x ¼ ^z ^y ¼ 0 Á Á Á ð1:27Þ ^ x ^x ¼ ^y ^y ¼ ^z ^z ¼ 1 ð1:28Þ Á ð1:29Þ Therefore, Eq.

The normal to the plane is obtained through use of the vector product. The vector A is then deﬁned between a general point (x,y,z) and any of the points given. The form in (b) is found from (a): (a) Two vectors necessary to calculate the normal vector to the plane are P1 to P2: A ¼ ^ x ð3 À 1Þ þ ^y ð1 À 0Þ þ ^z ðÀ2 À 2Þ ¼ ^x 2 þ ^y 1 À ^z 4 P1 to P3: B ¼ ^ x ð2 À 1Þ þ ^y ð3 À 0Þ þ ^z ð2 À 2Þ ¼ ^x 1 þ ^y 3 These two vectors are in the plane. Therefore, the normal vector to the plane may be written as: n ¼ A Â B ¼ ð^x 2 þ ^ y 1 À ^z 4Þ Â ð^x 1 þ ^y 3Þ ¼ ^x 12 À ^y 4 þ ^z 5 A general vector in the plane may be written as: C ¼ ^x ðx À 1Þ þ ^y ðy À 0Þ þ ^z ðz À 2Þ where point P1 was used, arbitrarily.