National University

• Clearly demonstrate an ability to employ vector operations and properties to include additive, scalar, and vector cross product operations. Using such fundamental properties, the student will provide evidence of employing vectors to define lines and planes in 3-space, and will demonstrate competence in extending such capability to conic surfaces.
• Show ability to define vector-valued functions in terms of the scalar functions as components to the basis vectors. Interpret resulting-properties correctly. Successfully differentiate and integrate the scalar components.
• Provide evidence of understanding and competence in differentiation and integration of functions of more than one variable. The student will demonstrate sufficient capability with partial derivatives so as to permit derivations of directional derivatives and gradient functions. The student will be capable of developing the elements of Jacobean Matrix. Display capability to incorporate change of variable and other associated applications as specified in the corresponding goal of this syllabus. Comprehension of the classic theorems of mechanics will be evaluated by application to elementary surface.

Course Goals:
This course is an extension of the development of topics within Calculus1 & II to include 3-dimensional surfaces and their vector representations. Vector-valued position, velocity, and acceleration vectors are employed to describe arc length, curvature, and motion in a 3-space. Multiple integration of scalar functions in many variables, and vector analysis leading to classic results of physical mechanics concluded this course, and the calculus sequence.

A principal objective of the course is to solidify the student's appreciation of the underlying thoughts and historical development of the calculus, as well as a solid understanding of the power of its techniques. This course enlarges foundations in differentiation and integration, coupled with their associated application in 3-dimensional space using vector-valued functions and multiple integration.

To obtain this objective, the following course goals have been established in which the student shall:

• Develop the concept of vectors in 3-space from the intuitive as well as algebraic sense. Define properties of relationships between vectors to include scalar and vector products. Define lines and planes in space. Strengthen the student's intuitive appreciation of resulting vector operations as substantiated by associated algebraic computations.
• Be exposed to vector-valued functions using all algebraic and transcendental functions derived in the preceding calculus sequence. Differentiation and integration of vector-valued functions leading to velocity, and acceleration vectors will be developed together with their tangential and orthogonal properties. Applications to arc length and curvature will exploit the properties of parametric representations developed in Calculus II.
• Extend earlier techniques of integration and differentiation of scalar functions to the field of vector-valued functions. Partial differentiation will be introduced to permit development of directional derivatives and the concept of the gradient function. With these capabilities, the student will broaden concepts of extrema for functions of more than one variable, undertake double and triple integral, considerations for centers of mass, surface area, and associated applications. Introduction to independence of path for line integrals within conservative vector fields will conclude this course by exposing the student to classic results in the physical mechanics including Green's, Stoke's and the Divergence Theorem.

Course content.
This course is an extension of the development of topics within Calculus1 & II to include 3-dimensional surfaces and their vector representations. Vector-valued position, velocity, and acceleration vectors are employed to describe arc length, curvature, and motion in a 3-space. Multiple integration of scalar functions in many variables, and vector analysis leading to classic results of physical mechanics concluded this course.

Incomplete A grade given at the discretion of the instructor when a student who has completed at least two-thirds of the course class sessions and is unable to complete the requirements of the course because of uncontrollable and unforeseen circumstances. The student must convey these circumstances (preferably in writing) to the instructor prior to the final day of the course. If an instructor decides that an "Incomplete" is warranted, the instructor must convey the conditions for removal of the "Incomplete" to the student in writing. A copy must also be placed on file with the Office of the Registrar until the "Incomplete" is removed or the time limit for removal has passed. An "Incomplete" is not assigned when the only way the student could make up the work would be to attend a major portion of the class when next offered.

An "I" that is not removed within the stipulated time becomes an "F." No grade points are assigned. The "F" is calculated in the grade point average.

Withdrawal Signifies that a student has withdrawn from a course after beginning the third class session. Students who wish to withdraw must notify their admissions advisor before the beginning of the sixth class session in the case of graduate courses, or before the seventh class session in the case of undergraduate courses. Instructors are not authorized to issue a "W" grade.