MATH 2760 - Analytic Geometry & Calculus 3

Credit Hours: 4.00

Objectives: Students will:

1. Add and subtract vectors and be able to geometrically represent these operations.
2. Find a scalar multiple of a vector and be able to represent it graphically.
3. Calculate a dot product of two vectors.
4. Calculate the work done by a force as it moves an object.
5. Find the angle between two vectors.
6. Find the scalar projection and vector projection of one vector onto another.
7. Use projections to find the distances between: a point and a line, a point and a plane, two lines.
8. Calculate a cross product of vectors and demonstrate knowledge of its geometric interpretation.
9. Find the equation of a line in R2 and R3 in vector nonparametric form.
10. Find the symmetric equations of line in R3.
11. Find the equation of a plane in vector nonparametric form.
12. Be able to rewrite a vector equation in Cartesian nonparametric form.
13. Calculate a box product and demonstrate knowledge of its geometric interpretation.

Outcome 2: Upon completion of this course, the student will be able to perform calculus on vector valued functions and obtain measurements in geometry and kinematics.

Objectives: Students will:

1. Express the equation of a line in vector parametric form and scalar parametric form.
2. Differentiate a vector-valued function.
3. Find the velocity, speed and acceleration vectors of a particle in space given its position function.
4. Find the length of an arc described parametrically or as a vector-valued function.

Outcome 3: Upon completion of this course, the student will be able to recognize quadric surfaces and their equations, graph quadric surfaces, and obtain traces of quadric surfaces.

Objectives: Students will:

1. Recognize the equation of the following quadric surfaces: ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids.
2. Graph the following quadric surfaces: ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids.
3. Find the traces of a quadric surface.
4. Find the cross section of a quadric surface with a plane perpendicular to one of the coordinate axes.

Outcome 4: Upon completion of this course, the student will be able to find the domain and range, calculate limits, and perform differential calculus on functions of two or three variables, as well as to use differential calculus to analyze the shape of a graph and to solve optimization problems.

Objectives: Students will:

1. Graph a function of 2 variables and identify its domain and range.
2. Find the limit of a function of several variables.
3. Find the partial derivatives of a function of several variables and interpret the derivative geometrically.
4. Find the total differential of a function of several variables and use it in an application
5. Use the chain rule for function of several variables to find a partial derivative of a function.
6. Find the directional derivative of a function of two or three variables in any direction.
7. Find the gradient of a function in two or three variables.
8. Find the maximum value of the directional derivative and a unit vector in the direction of the maximum directional derivative at a given point for a function of two or three variables.
9. Find the equation of a line normal to and the line tangent to a level curve at a given point.
10. Find the equation of a line normal to and the plane tangent to a level surface at a given point.
11. Find critical points of a function in two or three variables.
12. Classify relative extrema of functions of two variables using the second derivative test.
13. Use second derivative test to find the optimum solution of an applied problem.
14. Use the method of Lagrange multipliers to solve constrained optimization problems.
15. Use the method of Lagrange multipliers to solve applied constrained optimization problems.

Outcome 5: Upon completion of this course, the student will be able to evaluate double and triple integrals in rectangular, polar, cylindrical, and spherical coordinates, and to apply these tools to calculate areas and volumes.

Objectives: Students will:

1. Evaluate an iterated integral.
2. Change the order of integration of an iterated integral.
3. Convert a double integral to an iterated integral.
4. Use a double integral to find the area of a region in the plane.
5. Use a double integral to find the volume of a solid.
6. Use polar coordinates to evaluate a double integral.
7. Use an iterated integral to evaluate a triple integral
8. Use a triple integral to find the volume of a solid.
9. Use cylindrical coordinates to evaluate a triple integral.
10. Use spherical coordinates to evaluate a triple integral.
11. Use a double integral to calculate the area of a surface.

Outcome 6: Upon completion of this course, the student will be able to perform calculus on vector fields, with topics including divergence, curl, line integrals, and surface integrals, as well as to manipulate integrals using the fundamental theorem, Green’s theorem, the divergence theorem, and Stokes’ theorem.

Objectives: Students will:

1. Evaluate a line integral over a curve described in scalar parametric form.
2. Use the dot product to evaluate a line integral on a curve given in scalar Cartesian form.
3. Use a line integral to calculate the work done by a force moving a particle along a curve.
4. Use Green’s theorem to evaluate an appropriate line integral.
5. Evaluate a line integral using the fundamental theorem.
6. Calculate the divergence and the curl of a vector field.
7. Find the area of a parametric surface.
8. Evaluate a surface integral.
9. Use Stokes’ theorem.
10. Use the divergence theorem.

• Communication: The graduate can communicate effectively for the intended purpose and audience.
• Critical Thinking: The graduate can make informed decisions after analyzing information or evidence related to the issue.
• Global Literacy: The graduate can analyze human behavior or experiences through cultural, social, political, or economic perspectives.
• Information Literacy: The graduate can responsibly use information gathered from a variety of formats in order to complete a task.
• Quantitative Reasoning: The graduate can apply quantitative methods or evidence to solve problems or make judgments.
• Scientific Literacy: The graduate can produce or interpret scientific information presented in a variety of formats.

1. Vectors
   1. Vector operations
      1. Addition/subtraction
      2. Scalar multiplication
      3. Dot Product
      4. Cross product
      5. Box product
   2. Analytic Geometry
      1. The angle between two vectors
      2. The projection of one vector onto another
      3. Equations of a line in both parametric and nonparametric forms
      4. Equations of a plane in both parametric and nonparametric forms
      5. Using the projection to find the distance between two objects in the plane or in space.
   3. Vector valued functions and parametric equations
      1. Graphs of parametric equations and vector valued functions
      2. Limits of vector valued functions
      3. Continuity of vector valued functions
      4. Differentiation of vector valued functions
      5. Velocity, speed and acceleration of a particle in space
      6. The length of an arc described parametrically or by a vector valued function
2. Surfaces in space
   1. Intercepts, traces, and cross sections
   2. Cylinders
   3. Quadric surfaces
   4. Graphs of functions of 2 variables
3. Functions of several variables
   1. Domains of functions of more than one variable
   2. Limits of functions of more than one variable
   3. Continuity of functions of 2 or 3 variables
   4. Partial derivatives
   5. Total Differential
   6. Chain rule for functions of several variables
   7. Directional derivatives and gradients
      1. Tangent Planes and normal lines
   8. Extrema for functions of 2 variables
   9. Lagrange multipliers
4. Multiple integration
   1. Iterated integrals
   2. Double integrals
      2. Areas and volumes
   3. Using polar coordinates to evaluate a double integral
   4. Triple integrals
      3. Volumes
   5. E Using spherical and cylindrical coordinates to evaluate a triple integral
   6. Surface area
5. Vector calculus
   1. Line integrals of conservative and non-conservative vector fields over closed and non-closed curves
   2. The fundamental theorem of line integrals
   3. Green’s Theorem
   4. Divergence and Curl
   5. Parametric surfaces and their areas
   6. Surface integrals
   7. Stokes’ theorem
   8. Divergence theorem