o 1 1 2 2 3 3 where u u1 u2 u3 and v v1 v2 v3 o cos o cos 1 Key Concepts in Math 230 Final Survival Sheet Chapter 12 Vectors and 3D Space The vector called u from point A Ax Ay Az to B Bx By Bz is 2 2 2 Equation of a sphere 2 0 2 0 2 0 2 Properties of Vectors o o o 0 0 0 0 o o o 1 2 1 2 o 1 o o 1 2 1 2 Unit Vectors o 1 0 0 o 0 1 0 o 0 0 1 Dot Products Properties of Dot Products o o o o 0 o 0 if and only if Cross Products scalar o det 1 2 3 1 2 3 o sin o Properties of Cross Products o o o vector if and only if the area of the parallelogram spanned by u and v v A u 2 Key Concepts in Math 230 Final Survival Sheet Triple Product o The triple product of vectors u v and w is o det 3 2 1 3 2 1 1 2 3 o the volume of the parallelepiped spanned by u v and w w v u o Lines o 0 The parametric line through r0 in the direction v o 1 0 1 0 1 The line segment from r0 to r1 o 1 0 1 The parametric line through r0 and r1 Quadratic Surfaces o General form 2 2 2 0 o Determining the shape see chart A B C quadratic terms All positive All negative Non zero different signs One is zero the other two have same sign One is zero the other two have different signs Chapter 13 Vector Functions Type Ellipsoid Ellipsoid Hyperboloid Paraboloid elliptic Example 4 2 2 2 1 4 2 2 2 1 4 2 2 2 1 2 2 0 Paraboloid hyperbolic 2 2 0 3 3 The unit tangent vector of a graph is Differentiation rules o o o o If If If where c is a constant If 3 Key Concepts in Math 230 Final Survival Sheet The length of the curve is 2 2 The curvature of at t0 is where T is the unit tangent vector Velocity Acceleration Chapter 14 Partial Derivatives The partial derivative of f x y is a derivative taken with one variable treated as a constant if where c is a constant then o Notation The level curves of f x y are of the form where c is a The level surfaces of f x y z are of the form where c is constant a constant The linear approximation of f x y at x0 y0 is 0 0 0 0 0 0 0 0 The gradient of f x y z is The tangent plane of f x y z at the point x0 y0 z0 is 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The directional derivative of f x y z is 1 o The directional derivative is at a maximum when In that case Clairout s theorem 2 2 or The Laplace Equation 0 The Heat Equation 2 0 The Wave Equation 2 0 Chain Rule o If then Local minima and maxima o Occur at critical points x0 y0 is a critical point of f x y if 1 0 0 0 or 2 fx or fy is not defined at x0 y0 4 Key Concepts in Math 230 Final Survival Sheet o Classify critical points with the determinant of the Hessian 2 If D 0 and fxx 0 then that point is a local minimum If D 0 and fxx 0 then that point is a local maximum If D 0 then that point is a saddle point How to find the global absolute maximum or minimum in on the region R 1 Find all the critical points on the interior of R 2 Find all the local minimums and maximums on the boundary of R To do this step a Use a parametric representation of the boundary like then find the minimum and maximum of f or b Use Lagrange multipliers i If the boundary has the form g x y c then the local minimums and maximums of f on the boundary obey for some real number 3 Evaluate the function at all of the points found in steps 1 and 2 to get the absolute maximum and minimum Chapter 15 Multiple Integrals Fubini s theorem If the region of integration D is type I then If D is type II then In polar Center of mass of a thin plate 1 1 5 Key Concepts in Math 230 Final Survival Sheet Surface area of z x y Triple integrals 1 2 2 Limit rules and order of integration rules are similar to those used with double integrals Center of Mass of a three dimensional object 1 1 1 Spherical coordinates o the distance from the origin o the angle from the positive z axis o the angle of revolution around the z axis 2 2 2 2 sin sin cos cos sin sin tan Triple integrals with spherical coordinates sin cos sin sin cos 2 sin Chapter 16 Vector Calculus Vector fields o 3 o 3 3 Conservative fields o A vector field F is conservative if for some scalar function f x y z o A vector field is conservative if and only if the curl o If the field is conservative then the fundamental theorem of line integrals works 6 Key Concepts in Math 230 Final Survival Sheet Curl o In 3D if curl det o In 2D curl The fundamental theorem of line integrals end start Two types of line integrals o General form o The Work Integral Green s theorem 2D o If C is a closed loop then Stokes theorem 3D o If C is a closed loop then where curl where and 7 Key Concepts in Math 230 Final Survival Sheet Surface integrals Surface area 1 2 2 1 If and 1 where o If and 1 with upward orientation then 1 where Divergence theorem div Flux