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# Calculus 3 MATH 311

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This 48 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 311 at Millersville University of Pennsylvania taught by J. Buchanan in Fall. Since its upload, it has received 47 views. For similar materials see /class/223528/math-311-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.

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Date Created: 10/15/15

The Cross Product MATH 311 Calculus III J Robert Buchanan Department of Mathematics Fall 2007 Introduction Recall the dot product of two vectors is a scalar There is another binary operation on vectors called the cross product whose result is yet another vector Introduction Recall the dot product of two vectors is a scalar There is another binary operation on vectors called the cross product whose result is yet another vector Definition The determinant of a 2 x 2 matrix of real numbers is defined by a1 a2 b1 b2 ab 7 a2b1 VFinthihe following 2 x 2 determinants 1 2 o 3 1 2 71 75 4 30 42 3 x 3 Determinants Definition The determinant of a 3 x 3 matrix of real numbers is defined as a combination of three 2 x 2 determinants as follows a1 a2 a3 b1 b2 b3 a1 C1 C2 C3 b2 b3 C2 03 b1 b3 a C1c33 b1 b2 C1 C2 3 x 3 Determinants Definition iThe determinant of a 3 x 3 matrix of real numbers is defined 7 as a combination of three 2 x 2 determinants as follows a1 a2 a3 b1 b2 b3 a1 C1 C2 C3 b2 b3 C2 C3 b1 b3 C1 C3 83 b1 b2 C1 C2 Remark this method of calculating the 3 x 3 determinant is called expansion along the first row Find the following 3 x 3 determinants 1 2 71 0 3 O 1 75 4 2 3 72 2 e 1 1 5 72 1 6 Cross Product Definition For two vectors a 3132613 and b b1 b2 b3 we define the cross product or vector product of a and b to be i j ax b a1 32 33 b1 b2 b3 6392 33 b2 b3 Cross Product Definition For two vectors a 3132613 and b b1 b2 b3 we define the cross product or vector product of a and b to be i j ax b a1 32 33 b1 b2 b3 6392 33 b2 b3 Note the cross product is another vector in V3 Ea 13 4 and b lt2 7 75gt then evaluate the following 9 a x b 9 b x a 9 a x 0 perties ofhe ros Product Foranyvectorain V3 ax aOanda x 0 0 Properties of the Cross Product Foranyvectorain V3 ax aOanda x 00 ther Product i Foranyvectorain V3 ax aOanda x 00 For any vectors a and b in V3 a x b is orthogonal to both a and b rqurties he Cross Product i Foranyvectorain V3 ax aOandax 00 Cross Product Algebra For any vectors a b and c in V3 and any scalar d the following hold 0 a x b 7b x a anticommutativity 9 dax bdaxbax db 9 a x b c a x b a x c distributive laW 9 a b x c a x c b x c distributive laW 9 a b x c a x b c scalar triple product 9 a x b x c a cb 7 a bc vector triple product Goetric Interpretation 7 The vector a x b is perpendicularto the plane in R3 containing both a and b Geometric Interpretation 7 The vector a x b is perpendicularto the plane in R3 containing both a and b Use the righthand rule to determine the direction in which a x b points Geometric Interpretation The vector a x b is perpendicularto the plane in R3 containing both a and b Use the righthand rule to determine the direction in which a x b points For nonzero vectors a and b in V3 if 0 is the angle between a and b Where 0 g 0 7r then Ha X bii M M Sin 0 7 eometric Interpretation The vector a x b is perpendicularto the plane in R3 containing both a and b Use the righthand rule to determine the direction in which a x b points For nonzero vectors a and b in V3 if 0 is the angle between a and b Where 0 g 0 7r then Ha x W HaH HbHsiW Geometric Interpretation cont Two nonzero vectors a and b in V3 are parallel if and only if a x b 0 eomc Interpretaon cnt Two nonzero vectors a and b in V3 are parallel if and only if a x b 0 Area of a Parallelogram Area basealtitude HbH HaH sin0 Ha x bH iFind the area ofthe parallelogram with two adjacent sides formed by the vectors a 32 2 and b lt5 46 Distance from a Point to a Line 7 if 7 7 KO l 4 1 HPQHsme 1 R HITCS x Wm T HPRH 7 dHTousino iFind the distance from the point with coordinates 211 tothe line through the points with coordinates 7213 and 3 71 72 Volume of a Parallelepiped Ha X W Volume Area of basealtitude Ha X b compaxbc Haxb m caxb iFind the volume of the parallelepiped with three adjacent sides formed by the vectors a 32 2 b 546 and c lt72 3 4 Consider a wrench being used to tighten a bolt A force vector F is applied at a point specified by a position vector r The torque vector is defined to be 7 r x F The spinning motion of a ball creates a force called the Magnus force w angular velocity ofthe spin radians per second spin vector 5 vector parallel to the axis of spin with magnitude to v velocity vector of the ball m CS x v where c gt O is constant Magnus Fo cec Fmcsgtltv mauus macs mus macs WEIGHT Homework 0 Read Section 104 0 Pages 825 828 1 67 odd Stokes Theorem MATH 311 Calculus II J Robert Buchanan Department of Mathematics Fall 2008 Backg ound i Recall the vector form of Green39s Theorem FdrRVXFrde where C is a piecewise smooth positively oriented simple closed curve in the xyplane enclosing region R and the vector field FX y ltMXy NXy0gt Today we will generalize this result to three dimensions fit Stokes Theorem Theorem Stokes39 Theorem Suppose that S is an oriented piecewisesmooth surface with unit normal vector n bounded by the simple closed piecewisesmooth boundary curve 88 having positive orientation Let FX7 y7 2 be a vector field Whose components have continuous first partial derivatives in some open region containing 8 Then ngFxyzdrSVXFnds Interpretation Sometimes Stokes39 Theorem is written as gsFXy72 drSVx Fnd3jgsF Tds where T is the unit tangent in the direction of 88 Interpretation The line integral of the tangential component of F is equal to the flux of the curl of F This integral is the average tendency of the flow of F to rotate around path 88 Evaluate F dr where FXy z iyzi X 22k and 88 as is the intersection of the plane y z 2 and the cylinder X2 y2 1 Evaluate V gtlt F ndS where FX7y z yzi X2 xyk s and S is the part of the sphere X2 y2 z2 4 that lies inside the cylinder X2 y2 1 and above the xyplane quot V I r ntepretation of the Curl 1 of 3 Let P X0y0 20 be any point in a vector field F and let 8 be a circular disk of radius a gt O centered at P Let Ca be the boundary of 8a Intrretation of the Curl 2 of 3 fiFdr SVXFndS waz Kw Fn1pa for some point Pa 6 8 by the Integral Mean Value Theorem ntepretation of the Curl 3 of 3 1 VxFntPa E apdr 1 a39L39B VXF quot Pa aln8 fdr Vth nttp If F describes the flow of a fluid then F t T ds is the c circulation around C the average tendency of the fluid to circulate around the curve O V x F nlp attains its maximum when V x F is parallel to n 0 We can define rotF V x F n o F is an irrotational vector field if and only if V x F 0 Suppose F lt3y 42 76Xgt Find the direction of the maximum value of V x F n ronal Vector Fl Suppose that FX y 2 has continuous partial derivatives throughout a simply connected region D then V x F 0 in D if and only if fl F dr O for every simple closed curve C in D c 0 Suppose V x F 0 roatinal Vector Fiel Suppose that FX y 2 has continuous partial derivatives throughout a simply connected region D then V x F 0 in D if and only if fl F dr O for every simple closed curve C in D c 0 Suppose V x F 0 0 Suppose fl F dr O for every simple closed curve C in c Inclusive Result Suppose that FX7 y7 2 has continuous first partial derivatives throughout a simply connected region D then the following statements are equivalent 0 F is conservative in D ie F Vf 9 Fdr is independentofpath in D c 9 F is irrotational in D ie V x F 0 in D 9 fl F dr O for every simple closed curve in D c Let FX7y7 z y2i 2xy e32j 3ye32k and show that F is conservative Homework 0 Read Section 148 0 Pages 12071208 1 33 odd

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