Multivariable Calculus ImportantNotesChapter 11: Geometry in Space and Vectors11.1 Cartesian Coordinates in Three-SpaceDistance Formula in Three-SpaceThe distance formula can be generalized to three dimensions (or three-space) by consideringa cube diagonal instead of a square diagonal. The distance fromP1toP2in three-space isgiven by|P1P2|=p(x2-x1)2+ (y2-y1)2+ (y2-y1)2Equation of a Sphere in Three-SpaceThe equation of a circle extended into three-space translates into the equation of a spherewhen the z coordinate is taken into account. For a sphere centered at (h,k,l) the followingequation holds(x-h)2+ (y-k)2+ (z-l)2=r2Expansion of this formula leads to the followingStandard Equation of a Sphere in Three-Spacex2+y2+z2+Gx+Hy+Iz+J= 0Arc Length in Three-SpaceFor a function of t parameterized byx=f(t)y=g(t)z=h(t)atbThe derivatives essentially represent infinitesimal change, using this fact, the distance for-mula and integrating yields the followingL=Zbaq[f0(t)]2+ [g0(t)]2+ [h0(t)]2dtWhere L is the arc length of the parameterized function on the specified interval [a, b].Equation of a General Plane in Three-SpaceA plane in three-space represents a similar concept as a line does in two-space.A plane is defined generally in three-space asA(x) +B(y) +C(z) =DA plane is defined by its normal vector~nor the vector that is perpendicular to the plane ata point.WherehA, B, Ciis Normal to the Plane and defined as~n1
11.2 VectorsOften it is easier to do multivariable calculus if a function is dealt with as a vector valuedfunction.Prior to doing calculus in vectors one must learn the properties of vectors and how to ma-nipulate them properly.Magnitude of a three component vectorThe magnitude of a three component vector coincides to the distance formula inthree-space where~u=u1ˆi+u2ˆj+u3ˆkis given byk~uk=p(u1)2+ (u2)2+ (u3)2Definition of a unit vector (vector of length one)Often it is easier to manipulate vectors in a standard form of length one, this is easilyachieved by dividing a vector by its own magnitude shown as~uunit=~uk~ukTension ProblemsTension problems are in which a weight is suspended by multiple wires or ropes. To solvethese find one of the variables in terms of the other(s) and substitute this value into therespective equation and solve.