Lecture 7 Multivariable functions not on rst exam Visualizing level curves f x y x y g x y x y h x y x x y Domain function Major topics on Exam 1 Vectors Dot product Cross Product Set of numbers where the function is de ned addition scalar multiplication nding lengths angles testing for perpendicularity determinants areas of triangles and parallelograms nding perpendicular vectors equations of planes Equations of planes Find a plane that contains 1 0 0 0 1 0 0 0 1 Vectors 1 1 0 1 0 1 are both in the plane Now do the cross product with these two vectors to get a vector a b c The equation for the plane is ax by cz d If the matrix is a square and the determinant of that matrix is not zero three planes determine a point x y z 1 x y z 0 two planes determine a line Matrices Matrix multiplications not commutative Linear systems Inverse matrix Geometry Parametric equations Parametric curves Inverse of a 3x3 matrix Compute matrix of minors Flip signs Take the transpose Divide by the determinant Parametric curves and projections 12 6 25
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