MATH 2840 – CALCULUS III

OBJECTIVES

Chapter 13:

1. Understand the three dimensional rectangular coordinate system (under the right hand rule), including points, planes, and surfaces in 3-space.

2. Be able to find the distance between two points in 3-space.

3. Be able to write the equations for a sphere in 3-space.

4. Understand the definition of a vector and position vectors in 2- and 3-space. Know the vector properties and be able to find the magnitude and direction of vectors and unit vectors.

5. Be able to add and subtract vectors, multiply vectors by scalars, and find dot products and cross products of vectors. Understand the geometric meaning of the results of these operations.

6. Be able to find the angle between two vectors and the direction angles and direction cosines of a vector. Be able to determine whether two vectors are parallel or orthogonal.

7. Be able to find the scalar and vector projections of one vector onto another.

8. Be able to use vectors and vector operations in real world applications problems.

9. Understand and be able to find the parametric equation for a line. Be able to show if two lines are parallel, skew, or intersecting. If intersecting, be able to find the point of intersection.

10. Understand and be able to find the vector and scalar equations for a plane in 3-space. Be able to determine whether two planes are parallel and, if not, determine the angle between them and the line of intersection.

11. Understand the difference between cylinders and quadric surfaces. Recognize the conic cylinders and the common quadric surfaces and, given their equations, be able to sketch them using traces in appropriate planes.

Chapter 14:

1. Understand the definition of a vector (valued) function. Be able to find the limit and the domain of vector functions.

2. Be able to determine whether a vector function is continuous and find the parametric equations and sketch the associated space curve.

3. Be able to find the derivatives and integrals of vector functions. Be able to use the derivative to find tangent lines and tangent vectors to space curves.

4. Know the rules for differentiation of vector functions.

5. Be able to find arc length and curvature of a vector function or space curve.

6. Understand and be able to find the normal and binormal vectors of space curves, and the normal planes.

7. Be able to find velocity and acceleration vectors, and the speed of a particle moving along a space curve, and the tangential and normal components of the acceleration vector.

8. Be able to use vector functions in real world application problems.

Chapter 15:

1. Understand the definition of a function in several variables and be able to find the domain and range of the function.

2. Be able to sketch the level curves of a function of two variables, f(x,y), and the level surfaces of a function of three variables, f(x,y,z).

3. Understand the definition of limit for a multivariable function. Be able to determine a limit or show that a limit does not exist by calculating limits on different paths.

4. Understand what it means for a function to be continuous and be able to determine whether a function of two or three variables is continuous.

5. Be able to find partial derivatives of a function of several variables and understand the geometric interpretation. Know Clairaut’s Theorem.

6. Be able to use partial derivatives of functions of several variables in real world application problems.

7. Be able to find the tangent planes to surfaces in 3-space. Know the significance of the normal vector to a plane and be able to find it.

8. Understand linearization as an approximation of tangent planes and be able to find linear approximations. Be able to find the total differential of a function in two or more variables.

9. Understand and be able to use the chain rule to compute derivatives of composite functions in multiple variables. Be able to find derivatives implicitly.

10. Understand the definition of the directional derivative and gradient and be able to compute them. Be able to use directional derivatives to find the maximum rate of change.

11. Understand the definition and be able to find local and absolute maximum and minimum values and saddle points of functions of two variables. Be able to find critical points.

12. Know and be able to use the Second Derivative Test to determine whether a critical point is a local maximum, local minimum, or saddle point.

13. Be able to use Legrange multipliers to find maximum and minimum values of functions of two or three variables subject to one or more constraints.

Chapter 16:

1. Understand the definition of double and triple integrals as the limit of Riemann sums.

2. Be able to set up and evaluate double integrals in rectangular and polar coordinates. Be able to convert from rectangular to polar coordinates and vice versa.

3. Know Fubini’s Theorem for double and triple integrals.

4. Be able to set up and evaluate triple integrals in rectangular, cylindrical, and spherical coordinates. Be able to convert from rectangular to either cylindrical or spherical coordinates and vice versa.

5. Be able to find area and volume using double or triple integrals, as appropriate. Be able to switch limits of integration to facilitate evaluating integrals.

6. Be able to solve real world applications problems using double and triple integrals, to include finding moments, mass, centers of mass, and moments of inertia.

7. Be able to use a Jacobian to evaluate a change of variables in a double integral.

Chapter 17:

1. Understand the definition of a vector field in 2- and 3-space. Be able to sketch a vector field F.

2. Be able to find and sketch a gradient vector field.

3. Understand the definition of a line integral and be able to find the line integral of a function on a smooth plane or space curve. Be able to find line integrals of vector fields along smooth (or piecewise smooth) curves.

4. Be able to use line integrals to solve real world applications problems, to include finding work.

5. Understand and be able to apply the Fundamental Theorem for Line Integrals.

6. Understand what is meant by independent of path, and be able to determine when a line integral is independent of path.

7. Understand the definition of conservative and be able to determine whether a vector field is conservative. If a vector field is conservative, be able to find the potential function, f.

8. Know when a curve is closed and/or simple. Know when a region is open, closed, neither, and/or simply connected.

9. Understand and be able to apply Green’s Theorem.

10. Understand and be able to find curl and divergence of a vector field.

11. Know what a parametric surface is and how to write its vector equation. Be able to

find the equation of the tangent plane to a parametric surface at a given point.

12. Be able to calculate the surface area of a surface, S, whether S is given

parametrically or explicitly. Be able to set up and evaluate the surface integral of f

over the surface S.