Multi_Notes First Release - Multivariable Calculus Important Notes Chapter 11 Geometry in Space and Vectors 11.1 Cartesian Coordinates in Three-Space

Multi_Notes First Release - Multivariable Calculus...

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Multivariable Calculus Important Notes Chapter 11: Geometry in Space and Vectors 11.1 Cartesian Coordinates in Three-Space Distance Formula in Three-Space The distance formula can be generalized to three dimensions (or three-space) by considering a cube diagonal instead of a square diagonal. The distance from P 1 to P 2 in three-space is given by | P 1 P 2 | = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( y 2 - y 1 ) 2 Equation of a Sphere in Three-Space The equation of a circle extended into three-space translates into the equation of a sphere when the z coordinate is taken into account. For a sphere centered at (h,k,l) the following equation holds ( x - h ) 2 + ( y - k ) 2 + ( z - l ) 2 = r 2 Expansion of this formula leads to the following Standard Equation of a Sphere in Three-Space x 2 + y 2 + z 2 + Gx + Hy + Iz + J = 0 Arc Length in Three-Space For a function of t parameterized by x = f ( t ) y = g ( t ) z = h ( t ) a t b The derivatives essentially represent infinitesimal change, using this fact, the distance for- mula and integrating yields the following L = Z b a q [ f 0 ( t )] 2 + [ g 0 ( t )] 2 + [ h 0 ( t )] 2 dt Where L is the arc length of the parameterized function on the specified interval [ a, b ]. Equation of a General Plane in Three-Space A plane in three-space represents a similar concept as a line does in two-space. A plane is defined generally in three-space as A ( x ) + B ( y ) + C ( z ) = D A plane is defined by its normal vector ~ n or the vector that is perpendicular to the plane at a point. Where h A, B, C i is Normal to the Plane and defined as ~ n 1
11.2 Vectors Often it is easier to do multivariable calculus if a function is dealt with as a vector valued function. 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 vector The magnitude of a three component vector coincides to the distance formula in three-space where ~ u = u 1 ˆ i + u 2 ˆ j + u 3 ˆ k is given by k ~ u k = p ( u 1 ) 2 + ( u 2 ) 2 + ( u 3 ) 2 Definition of a unit vector (vector of length one) Often it is easier to manipulate vectors in a standard form of length one, this is easily achieved by dividing a vector by its own magnitude shown as ~ u unit = ~ u k ~ u k Tension Problems Tension problems are in which a weight is suspended by multiple wires or ropes. To solve these find one of the variables in terms of the other(s) and substitute this value into the respective equation and solve.

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