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MATH130 MULTIVARIABLE AND VECTOR CALCULUS

Course Objectives

At the end of this course the students are expected to;

•  Handle vectors fluently in solving problems involving the geometry of lines, curves, planes, and surfaces in space.
•  Examine functions of several variables, define and compute limits of functions at points and define and determine continuity
•  Define and compute partial derivatives, directional derivatives and differentials of multivariable functions and examine conditions of  differentiability; find th equation of the tangent plane to a surface at a point.
•  Find local extreme values of functions of several variables, test for saddle points, examine the conditions for the existence of absolute extreme values, solve constraint problems using Lagrange multipliers, and solve related application problems.
•  Use Rectangular, Cylindrical and Spherical Coordinates Systems to define space curves and surfaces in Cartesian and Parametric forms
•  Integrate functions of several variables
•  Examine vector fields and define and evaluate line integrals using the Fundamental Theorem of line integrals and Green’s Theorem; compute arc length
•  Define and compute the Curl and Divergence of vector fields and apply Green’s Theorem to evaluate line integrals, surface integrals and flux integrals.
•   Compute surface integrals.
•   Have an understanding of important theorems of vector calculus such as the divergence theorem in 3-space, Stokes’ theorem and the Fundamental Theorem of Vector Calculus.

Course Content

Vectors and analytic geometry in 3-space. Functions of several variables: limits, continuity, partial derivatives. Chain rule. Directional derivatives. Tangent planes and linear approximations. Extreme values. Lagrange multipliers. Double integrals, Double integrals in polar coordinates. General change of variables in double integrals. Surface parametrization and surface area in double integrals. Triple integrals in Cartesian, cylindrical and spherical coordinates. Parametrization of space curves. Line integrals. Path independence. Green's theorem in the plane. Surfaces and surface integrals. Flux integrals; gradient, divergence and curl. The divergence theorem in 3-space, Stokes` theorem, Fundamental Theorem of Vector Calculus.

Course Learning Outcomes

Student, who passed the course satisfactorily will be able to:

• Effectively write mathematical solutions in a clear and concise manner.
• Graphically and analytically synthesize and apply multivariable and vector-valued functions and their derivatives, using correct notation and mathematical precision.
• Use double, triple and line, surface integrals in applications, including Green's Theorem and Stoke's Theorem.
• Synthesize the key concepts of differential, integral and multivariate calculus.