Math Tools / Math Review

1. Math Tools / Math Review

2. A A + B q B Resultant Vector Problem: We wish to find the vector sum of vectors A and B Pictorially, this is shown in the figure on the right. Mathematically, we want to break the vector into x, y, and maybe z components and find the resultant vector

3. System of equations • Let 6x+7y=15 • And 4x-3y=9 • Now find x and y which satisfy these equations • I use “method of minors”

4. System of Equations cont’d The solution for x is found by creating a “minor” wherein the constants in the equation are substituted in place of the “x” value and the value of the minor is found. It is then divided by the value of the determinant. I then “back-substitute” the value of x into my initial equation and solve for y. 6(2.347)+7y=15 and solve for y (y=.1304)

5. Or I could use the minors again

6. Larger Systems Larger systems are broken down into their resultant minors. For example: 6x + 7y +10z =12 -9x+15y+2z=60 5x +12y-10z=15

7. Spherical Coordinates Cartesian coordinates: x, y, z Spherical coordinates: r, q, f Math Majors NOTE Theta!

8. Cylindrical Coordinates Cartesian coordinates: x, y, z Spherical coordinates: r, f, z Math Majors NOTE Phi!

9. Showing my age • In the old days, I would tell you to use your integral tables • Now, I say use your calculators to integrate IF YOU DARE!* * I only say this since I have seen some integrals which are easily found in the tables being integrated incorrectly by the calculator.

10. Partial Derivatives • So what is the difference between “d” and ? • “d” like d/dx means the function only contains the variable x. • When the function contains not just x but may be y and z, we use the partial differential, Note that the variables y and z are held constant when the differential operator acts on the function What is the solution to?

11. Introduction to “Del” • We can now make a special differential operator called “del”. Del is defined as • We treat “del” as a vector and thus, we can apply the “dot” and “cross” products to them. • But first, let’s recall the “dot” and “cross” product

12. The “dot” or scalar product • The scalar product is defined as the multiplication of two vectors in such a way that result is a vector q is the angle between A and B

13. “Cross” or vector product • The vector product is the multiplication of two vectors such that the result is a vector and furthermore, the resulting vector is perpendicular to the either of the two original vectors • The best way to find a vector product is to set it up as a determinant as shown on the right • q is the angle between A and B

14. First application of “del”: gradient • The gradient is defined as the shortest or steepest path up a mountain or down into a valley. • Let’s go back to f=xyz then • You see that “grad(f)” makes a vector which points in a particular direction. • Also, note that grad(f) takes a scalar function and makes a vector of it • A particle which travels through a region of space wherein the potential energy, U(x,y,z), varies as a function of space has a force exerted on it equivalent to

15. The scalar product and  • We can apply  to the scalar product i.e. • ·A where A is some vector • ·A is called the “divergence” of A or “div(A)”. • Geometrically, we are discussing if A is diverging from some central point. A is not diverging from a central point so Div(A) is equal to zero A is diverging from a central point so Div(A) is equal to some value

16. The vector product and  • We can apply  to the vector product i.e. • xA where A is some vector • xA is called the “curl” of A or “curl(A)”. • Geometrically, we are discussing if A is curling around some central point. A is curling around a central point so curl(A) is equal to some value A is not curling around a central point so curl(A) is equal to zero.

17. What about A · and A x? • These two products do not describe the geometrical properties • A · is not equal to  ·A due to the nature of the differential operator • -(A ·)U would be equivalent to A ·F, where F is a force described by -U • Likewise for -(A x)U

18. Two Special Integrals • Integrating over a closed loop: • Integrating over a closed surface:

19. A B D C Integrating over a closed loop • The loop can be circular or rectangular. From 0 to 2p Looping from Point A to Point D using straight line segments

20. Closed Surface Integral The vector n-hat is normal to the surface. This means that “da” must consist of the differential distance in the phi direction multiplied by the differential distance in the theta direction so Theta is integrated from 0 to p and phi is integrated from 0 to 2p Therefore if E depends only on R, then