1. Consider the line integral of the vector field F(x,y) = (x+y,x) over the portion of the parabola y = x^2 from origin to (1,1). Explain the steps you take to evaluate this line integral. Also find the result of the integral.

2. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Intuitively, what does a line integral measure?

2. What is a circulation, what does it measure?

3. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. What is a vector field? Give an example.

2. What is a gradient field? Give an example.

3. What is a flow line of a vector field. Give an informal definition in terms of fluid flow (e.g, water flow)

4. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Find the point(s) where the paraboloid z = x^2 + y^2 and the line x = 2+t, y = 2-t, z = 8t intersect.

2. If r(t) is the position vector of a moving object, how do you find its velocity and acceleration vectors?

3. How is the length of a curve related to its velocity?

4. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Express the 1/8 of the unit sphere in the first octant in spherical coordinates (use "rho", "phi" and "theta" for the spherical variables)

2. What surface does the equation phi = pi/4 describe in spherical coordinates?

3. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Express the upper half of the unit circle (centered at origin) in polar coordinates.

2. The infinitessimal area dA in rectangular coordinates is given by dx*dy. What is the corresponding representation for dA in polar coordinates?

3. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. If we know the density of a 3D object as a function of the position in 3D space over a region R, (say D(x,y,z)), how can we compute its total mass.

2. How does one use a triple integral to find volumes of 3D regions?

3. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. What is the significance of re-interpreting the integral of a function of two variables over a region as an iterated integral? (That is, why would we want to do this?)
2. How do we decide on the order in which to consider the variables in an iterated integral?
3.Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. In first semester calculus, if we had a function defined on an interval, we divided the interval into subintervals and computed the area of a rectangle corresponding to each subinterval. The sum of the areas of these rectangles approximated the integral of the one-variable function over the interval in question. Give an analogous description of an approximation to a function of two variables over a rectangular region.

2. If the region of interest (as discussed in question 1) is not rectangular, how do we approximate the integral of the function over the region?

3. When the integrand is positive over the interval, the regular integral of a one variable function gives the area of the region btw the function and the x-axis. What is the geometric meaning of a double integral when the integrand is positive?

4. Is there a way of getting the area of the region of integration in a double integral?

5. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. What is the meaning of parameter "lambda" in the method Lagrange multiplier?

2. What is a way of converting a constrained optimization problem to an unconstarined problem?

1. The definition given for differentiability for a function f(x,y) refers to a linear function L(x,y) and an error function E(x,y). How would we connect each of these with the intuitive idea we have discussed before of "zooming in" on a surface until we see something that looks like a plane?

2. What is the relationship between continuity and differentiability for functions of two variables?

3.Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Suppose we know that mixed partial derivatives of a function are continuous and f_xy (x,y) = x^2*sin(xy). what conclusion can you make?

2. Suppose f_xx (1,2) < 0. What does this tell you about the graph of the function f ?

3. Was there anything that was confusing to you in this section? Is there any specific or general questions you would like to have answered?

1. Let z = x*y, x = ln(t), y = 1/t. Compute dz/dt using the chain rule. Show your steps.

2. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. True or False: Partial derivatives are a special case directional derivatie.

2. Suppose the gradient of a function z = f(x,y) at a point (a,b) is (-1,2). Find the directional derivative of f at (a,b) in the direction of (3,4)

3. Given a function f and a point (a,b) what is the direction in which the directional derivative is greatest?

4. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1.Write a parametric equation of the line that goes through the point (2,1,3) and parallel to the line y = x onthe xy plane.

2. Find a parametrization of the circle of radius 3 in the xz plane centered at (1,0,1) , clockwise

3. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. What does it mean to say that a function of two variables is locally linear?

2. Find the equation of the tangent plane to z = 2*x*y^2 at (-1,-1)

3. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. How confident are you with the rules for differentiating functions that you learned in first semester calculus?

2. Find both partial derivatives for the function z = exp(x^2*cos(y)) (exp stands for the exponential function with base e. This is the notation used by many computer algebra systems including Maple)

3. Answer problem 4 on page 690 of the textbook.

4. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. What is the geometric meaning of f_x(a,b)?

2. Is f_x(a,b) a scalar or a vector?

3. True or False: If f x (a,b )>0, then the values of f decrease as we move in the negative x -direction near (a,b).

4. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. True or False: The cross product of two vectors is another vector.

2. True or False: Like dot product, the cross product is defined for vectors of any dimension

3. True or False:

4. Was there anything that was confusing to you in this section? Is there any specific (or general) question you would like to have answered?

1. True or False: The dot product of two vectors is another vector

2. Suppose you have two vectors say u and v. You are free to choose the direction for the vector v without changing its magnitude. How should you choose its directions so that the dot product of u and v is as large as possible (you cannot change anything about u). What is the largest possible value?

3. How do you check algebraically if given two vectors are perpendicular?

4. What is a normal vector to a plane?

5. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Experience shows that most people who take Calculus C have already seen this material at least in some form. (Doesn't mean they remember all of it!) Have you seen all or most of this material before? ( Don't worry if the answer is no, I just need to know whether the ideas are new or review for you. )

2. What are the two most attributes of a vector?

3. What is a unit vector?

4. Given the vector u = (1,1,1), find the unit vector that is in the opposite direction as u.

5. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Suppose that z=f(x,y) is a function of two variables. True or False: If the limit at (0,0) of the function along the cross section formed by the xz-plane and the limit at (0,0) of the function along the cross section formed by the yz-plane both exist and are equal, the limit of the function at (0,0) exists. Explain your answer.

2. When is a function if two variables not continuous at a point (a,b)?

1. Consider the values in the table below:

Compute the value that would have to go in the center square if the function values are to correspond to a linear function of two variables. Explain how you got your answer.
2. What is the formula for the linear function of two variables that matches the entries in the table given in question 1?

3. What does the contour diagram of a linear function of 2 variables look like?
4. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. Explain why the following statement is true: "the more closely spaced the contours, the steeper the terrain; the more widely spaced the contours, the flatter the terrain." The truth of this statement depends heavily on what convention about contour diagrams?

2. Algebraically, how are contour lines obtained?

3. Geometrically, how are countour lines are obtained from a given surface?

4. How does one obtain the surface from a given countour diagram?

5. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?

1. What kind of object is the graph of a function of 2 variables?

2. Describe the graph of z = 1-x^2-y^2

3. What is the shape of the graph of a linear function of 2 variables?

4. Describe the graph of y = x^2 in 3 dimensions

5. Was there a specific section that was confusing to you? Is there any specific (or general) question you would like to have answered?