MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Lecture 1 Thu Sept 6 2007 Handouts syllabus PS1 ashcards Goal of multivariable calculus tools to handle problems with several parameters functions of several variables has a direction and a length A It is represented by Vectors A vector notation A a directed line segment In a coordinate system it s expressed by components in space A a1 a2 a3 a1 a2 j a3 k Recall in space x axis points to the lower left y to the right z up Scalar multiplication Formula for length Showed picture of 3 2 1 and used ashcards to ask for its length Most students got the right answer 14 a2 a2 a2 by reducing to the Pythagorean theorem in the You can explain why A 1 2 3 and its projection to the xy plane then derived A from length plane Draw a picture showing A of projection Pythagorean theorem B by head to tail addition Draw a picture in a parallelogram showed how Vector addition A A addition works componentwise and it is true that the diagonals are A B and B 3 2j k on the displayed example A Dot product B a1 b1 a2 b2 a3 b3 a scalar not a vector De nition A B A B cos Theorem geometrically A A A 2 cos 0 A 2 is consistent with the de nition Explained the theorem as follows rst A B C A B Then the law of cosines gives C 2 Next consider a triangle with sides A 2 B 2 2 A B cos while we get A 2 C C A B A B A 2 B 2 2A B C Hence the theorem is a vector formulation of the law of cosines B A Applications 1 computing lengths and angles cos B A Example triangle in space with vertices P 1 0 0 Q 0 1 0 R 0 0 2 nd angle at P 1 PQ PR 1 1 0 1 0 2 cos 71 5 2 5 10 P Q P R Note the sign of dot product positive if angle less than 90 negative if angle more than 90 zero if perpendicular 2 detecting orthogonality Example what is the set of points where x 2y 3z 0 possible answers empty set a point a line a plane a sphere none of the above I don t know 1 2 3 Answer plane can see by hand but more geometrically use dot product call A OP x 2y 3z 0 A OP cos 0 2 A OP So we P x y z then A get the plane through O with normal vector A 1 2 18 02 Lecture 2 Fri Sept 7 2007 We ve seen two applications of dot product nding lengths angles and detecting orthogonality u A cos is the component A third one nding components of a vector If u is a unit vector A along the direction of u E g A component of A along x axis of A Example pendulum making an angle with vertical force weight of pendulum F pointing downwards then the physically important quantities are the components of F along tangential direction causes pendulum s motion and along normal direction causes string tension Area E g of a polygon in plane break into triangles Area of triangle 21 base height 1 2 A B sin 1 2 area of parallelogram Could get sin using dot product to compute cos and sin2 cos2 1 but it gives an ugly formula Instead reduce to complementary angle 2 A rotated 90 counterclockwise drew a picture Then area of parallelogram by considering A B sin A B cos A B A a1 a2 then what is A showed picture used ashcards Answer A a2 a1 Q if A explained on picture So area of parallelogram is b1 b2 a2 a1 a1 b2 a2 b1 a1 a2 a1 b2 a2 b1 Determinant De nition det A B b1 b2 a1 a2 area of parallelogram Geometrically b1 b2 is counterclockwise or clockwise from A The sign of 2D determinant has to do with whether B without details a1 a2 a3 b1 b2 b2 b3 b1 b3 b b b Determinant in space det A B C 1 2 3 a1 a a c2 c3 2 c1 c3 3 c1 c2 c1 c2 c3 B C volume of parallelepiped Referred to the notes for more about Geometrically det A determinants Cross product only for 2 vectors in space gives a vector not a scalar unlike dot product j k a2 a3 a1 a3 a1 a2 B a1 a2 a3 De nition A b2 b3 j b1 b3 k b1 b2 b1 b2 b3 the 3x3 determinant is a symbolic notation the actual formula is the expansion B area of space parallelogram with sides A B direction normal to Geometrically A and B the plane containing A How to decide between the two perpendicular directions right hand rule 1 extend right hand 2 curl ngers towards direction of B 3 thumb points in same direction as A B in direction of A Flashcard Question j answer k checked both by geometric description and by calculation C A n where n Triple product volume of parallelepiped area base height B B C B C So volume A B C det A B C The latter identity can also be checked directly using components
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