Final Exam

• Determine whether two given vectors are parallel or perpendicular (or neither).
• Compute the angle between two (nonzero) vectors.
• Given a vector, find a unit vector in the same direction.
• Compute the dot product and cross product of two vectors.
• Find the component of one vector in the direction of another.
• Find the projection of one vector in the direction of another.
• Find a vector that is perpendicular to two given vectors.
• Compute the area of a parallelogram determined by two vectors or by three points.
• Compute the volume of a parallelepiped deermined by three vectors or by four points.
• Determine whether three vectors lie in the same plane.
• Given two points in space, find the equation(s) for the line between them (i.e. parametric equations, equivalently a vector-valued function, and implicit equations, for example symmetric equations).
• Given a point and a direction vector, find the line through that point with the given direction vector.
• Find the equation of a plane through a given point and with a given normal vector.
• Find the equation of a plane through three given points.
• Given an equation for a plane, find a normal vector.
• Given two lines, determine whether they are parallel (or equal), skew, or intersect in a single point.
• Given two planes, compute the angle between them, and determine whether they are parallel (or equal) or intersect in a line (in this case, be able to find the equation of the line!).


• Given a vector-valued function (of one variable!), find the derivative.
• Use the product formula for dot and cross products, etc. to compute derivatives of a dot product or cross product of two vector-valued functions of t, a scalar-valued function times a vector-valued function, etc.
• Given the position function r(t) of a particle moving in space, compute its velocity, speed, acceleration, and scalar acceleration.
• Given the acceleration function and initial position and velocity, compute the position function r(t) of a particle.
• Compute the arclength of a space curve.
• Given a vector-valued function r(t), compute the unit tangent vector.
• Given a sphere described in words (for example, "the sphere with center (0,1,-1) and radius 3") find an equation for it.
• Given an equation of a sphere, find its center and radius.


• contour plots, graphing level curves
• Computing partial derivatives of first and second order
• Computing derivatives (partial and ordinary) by implicit differentiation
• Computing tangent plane to surfaces of the form z = f(x,y) using x-curves and y-curves (traces)
• Computing tangent plane to surfaces of the form g(x,y,z) = 0 using the gradient of g
• Using the gradient to compute approximate values of a given function f(x,y) near a given point (a,b)
• Computing derivatives via the multi-variable chain rule
• Computing gradients and directional derivatives
• Finding critical points of a function of several variables using the first derivative test
• Classifying critical points using the second derivative test
• Finding maxima/minima with constraints by Lagrange multiplier method, application to optimization problems


• Computation of double integrals over rectangular regions
• Computation of double integrals over "vertically simple" and "horizontally simple" regions (region between graphs of two functions)
• Reversing the order of integration
• Computation of area and volume via double integrals
• Change of variables in multiple integrals: coordinate transformations, definition of and how to compute the Jacobian determinant, how to set up an integral after a change of variables, how to compute
• A nice example: evaluation of the Gaussian integral, the integral of exp(-x2) from minus infinity to infinity (or from zero to infinity)
• how to compute mass of a "plane lamina."
• Centroid (center of mass) and how to compute it
• Triple integrals in rectangular coordinates
• Volume by triple integrals
• Polar coordinates, cylindrical coordinates, spherical coordinates, elliptical coordinates: how to describe regions in terms of them, how to change and compute with them
• The formulas for converting between rectangular (Cartesian) and polar, cylindrical, spherical, or elliptical coordinates


• Definition of and geometric meaning of vector fields
• How to interpret a graph of a vector field
• gradient vector field of a function
• curl and divergence of a vector field, how to compute them, their geometric meaning
• Definition and evaluation of a line integral with respect to arclength (ds)
• Definition and evaluation of a line integral of a vector field (F.dr)
• Work and how to calculate it with a line integral
• Independence of path: definition
• Independence of path is equivalent to the integrand F being a gradient
• Conservative vector fields
• How to tell when a vector field is conservative
• How to compute a potential function of a conservative vector field
• Statement of Green's Theorem
• Interpretation of Green's Theorem as a conservation law for circulation/rotation
• How to use Green's Theorem to compute line integrals
• How to use Green's Theorem to compute area (which we did in an example, the Folium of Descartes)
• Flux of a vector field in the plane across a curve
• The Divergence Theorem in the plane, a.k.a. the "vector form of Green's Theorem"
• Definition and evaluation of integral with respect to surface area (dS)
• Definition and evaluation of surface of the form F. ndS; its geometric meaning as flux
• Statement of Divergence Theorem
• Analogy of Green's Theorem and Divergence Theorem to Fundamental Theorem of Calculus in one variable
• Physical/geometric meaning of the Divergence Theorem as a conservation law
• Using the Divergence Theorem to compute surface integrals
• Applying the Divergence Theorem to an inverse-square vector field to show that flux is the same for all closed surfaces containing the origin
• Statement of Stokes's Theorem
• Physical/geometric meaning of Stokes's Theorem
• Using Stokes's Theorem to evaluate line integrals or surface integrals
• How to tell when a vector field is the curl of another vector field
• When a vector field Fis the curl of another vector field G, how to use the formula to find G (you do not need to memorize the formula).

• Do the homework! It is excellent preparation for the exam.
• Study the midterms!
• Do the "Mandatory problems" linked from the web page.
• Master the material! Knowing the basic skills listed above will not guarantee that you will effortlessly solve every exam problem. But if you know the basic skills and can do all the midterm and homework problems, you should do extremely well on the exam.
• Work hard! A large fraction of your grade is still to be decided. It's a chance to improve your grade significantly. And, I will be happy to give the entire class excellent grades if the performance on the exam is excellent!