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1. Fundamental notions of higher dimensions (Chapters 12 of the textbook)----

• Vector geometry and analytic geometry of R3

• Calculus of vector valued functions with applications to motion in space

• Functions of several variables, including graphs and level curves

2. Differential calculus in higher dimensions (Chapters 13 of the textbook)----

• Differentiation of functions of several variables, including partial derivatives, the chain rule, directional derivatives, gradients, and differentials

• Optimization problems for functions of more than one variable, including critical point analysis and Lagrange multipliers

3. Integral calculus in higher dimensions (Chapters 14 of the textbook)----

• Double integrals in both rectangular and polar coordinates

• Triple integrals in rectangular, cylindrical, and spherical coordinates

• Change of variable theorem

4. Vector calculus o Vector fields (Chapters 15 of the textbook)----

• Line integrals

• Conservative vector fields and the fundamental theorem of line integrals

• Green’s Theorem

• The student will develop the ability to reason geometrically in dimensions higher than two, including mastery of basic notions of vector geometry and equations of lines, planes, and spheres.

• The student will develop the ability to think geometrically about the graphs of functions of two variables, including the significance of partial derivatives, directional derivatives, and gradients.

• The student will learn to do the standard differential operations including partial derivatives and gradients, and how to use them to do simple optimization problems.

• The student will learn to set up and evaluate integrals in two and three variables, including working with any of the standard coordinate systems.

• The student will understand and correctly interpret the basic notions of vector calculus, including line integrals of vector fields and Green’s Theorem.