Time: 12:40 pm

Date: Wednesday 8/13

Place: 1234 EB





By the way make sure that you know the following informal brief survey that I think that everybody must know:

1) vectors (in 2D, in 3D), how to add two vectors, multiply vector by scalar, how to find dot product of two vectors. How to determine whether they are perpendicular or not. How to find projection of a vector. How to find cross product. Don't forget that cross product is the vector which is perpendicular to these two vectors. Magnitude of the vector. Unite vector. Direction.

2) How to find parametric equation of the line which goes through the given point and has given direction. How to find perpendicular (normal) vector to the plane. Intersections of the lines, planes. If you know the parametric equation of the particle how to find the length that it travels during the given time? How to find the parametric equation of the line that passes through the given two points A, B (Don't forget about our magic formula " r(t) = OA+t AB ). How to find the equation of the plane that passes through the given 3 points.

3) Partial derivatives. Do you understand how to differentiate function of three variables? Don't forget that other variables you treat as a constants. How to find derivative of the function given implicitly. If you have function of two variables, how to find the critical points, how to determine whether they are local minimum, local maximum or saddle points (look at the sign of the determinant of the matrix of second derivatives, and look at the sign of second derivative of the function itself). domain, range and level curves of the function. Equation of the plane. Meaning of the coefficients a,b,c, where a(x-x_0)+b(y-y_0)+c(z-z_0)=0. (this is the normal vector to the plane)

4) Double integrals. Do you understand how to sketch the region of integration if you know the limits of a double integral, or vice versa how to find the limits of integral if you know the region over which you take integral. How to switch the order of integration.

Line integrals of the vector fields (work along the curve!). When vector field is conservative (how to check it) how to find potential function, how to find integral over the path of conservative vector field (look on potential function)

5) Greens theorem (formula for the flux, formula for the circulation): just remember that line integral over the curve becomes double integral over the domain bounded by this curve

6) how to find the surface area when the graph (or surface) is given explicitly z=f(x,y). What is the domain of that double integral (it is a shadow of that surface on the plane XY). How to find the surface area when the surface is given parametrically r(u,v), what is the domain of that double integration. How to integrate the function over the surface.

7) Double integrals, polar coordinates, density of the double integral in polar coordinates, limits of integral.

8)Triple integrals, cylindrical coordinates, density in cylindrical coordinates, limits of cylindrical coordinates.

9) spherical coordinates, density in spherical coordinates, limits in spherical coordinates.

Just remember the following rules (for triple integrals): If the domain of integration looks like piece of ball , domain between two spheres, or ice-cream like domain then go to the spherical coordinate system (especially when you see the expression like x^2+y^2+z^2). If the domain looks like cylindrical domain, base of which looks like a disk, or ellipse then go to the cylindrical coordinate system. For the double integrals: if you integrate over the domain between two circles, over the disk , over the ellipse, then go to the polar coordinates (and don't forget about the density).

10) Stoke's theorem: this is the formula about when double integral over a surface of the cross product nabla and vector field times normal vector becomes line integral. Pay attention that in the line integral you integrate over the edge (let's say over the boundary) of the surface.

11) Divergence theorem! this simplifies outward flux over the 2D surface and it becomes tripple integral of the divergence of the vector field. Do you know how to compute divergence?



Ok, that's all I remember right now.