Math Tools / Math ReviewResultant VectorSystem of equationsSystem of Equations cont’dOr I could use the minors againLarger SystemsSpherical CoordinatesCylindrical CoordinatesShowing my agePartial DerivativesIntroduction to “Del”The “dot” or scalar product“Cross” or vector productFirst application of “del”: gradientThe scalar product and The vector product and What about A ·  and A x ?Two Special IntegralsIntegrating over a closed loopClosed Surface Integral1Math Tools / Math Review2Resultant VectorProblem: We wish to find the vector sum of vectors A and BPictorially, 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)126()67(tan)67()126(ˆ6ˆ12Andˆ7ˆ6Let22xyBAyxByxAABA + B3System of equationsLet 6x+7y=15And 4x-3y=9Now find x and y which satisfy these equationsI use “method of minors”462818)4)(7()3)(6(3476eDeterminat 4System of Equations cont’d347.246)9)(7()3)(15(det39715xThe 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)5Or I could use the minors again1304.046)15)(4()9)(6(det94156y6Larger Systemsdet1012152156010712343412515910105297101221561012521591076detxLarger systems are broken down into their resultant minors.For example:6x + 7y +10z =12-9x+15y+2z=605x +12y-10z=157Spherical CoordinatesCartesian coordinates: x, y, zSpherical coordinates: r,  , Math Majors NOTE Theta! rzxyzyxzyxrrzryrx1122221222costancossinsinsincos8Cylindrical CoordinatesCartesian coordinates: x, y, zSpherical coordinates: r, , zMath Majors NOTE Phi! zzxyyxyxrzzryrx 1222122tansincos9Showing my ageIn the old days, I would tell you to use your integral tablesNow, I say use your calculators to integrateIF 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.10Partial 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, yzxyzxxfxyzzyxfexampleFor)(),,(:Note that the variables y and z are held constant when the differential operator acts on the functionWhat is the solution to??)(?)( xyzzxyzy11Introduction to “Del”We can now make a special differential operator called “del”. Del is defined aszzyyxxˆˆˆ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” product12The “dot” or scalar productThe scalar product is defined as the multiplication of two vectors in such a way that result is a vector is the angle between A and BcosˆˆˆˆˆˆBABAorbababaBAThenzbybxbBandzayaxaAzzyyxxzyxzyx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sinˆ)(ˆ)(ˆ)(ˆˆˆBABAzbabaybabaxbabaBAbbbaaazyxBAxyyxxzzxyzzyzyxzyx is the angle between A and B14First 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 tozxyyxzxyzxyzfˆˆˆ)( zzUyyUxxUUFˆˆˆ15The 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 diverging from a central point soDiv(A) is equal to some valueA is not diverging from a central point soDiv(A) is equal to zero16The 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 not curling around a central point socurl(A) is equal to zero.A is curling around a central point so curl(A) is equal to some value17What 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 )U18Two Special IntegralsIntegrating over a closed loop:Integrating over a closed surface: sdB adB19Integrating over a closed loopThe loop can be circular or rectangular.)2( rBsdBFrom 0 to 2ABCDLooping from Point A to Point D using straight line segmentsˆrdsd ydsxd sydsxdssdDCBAˆˆˆˆ20Closed Surface IntegralddRRddRdasin))(sin(2 24 REadE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