As we have seen, parametrizing surfaces and determining the orientation induced by a parametrization can be...

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• As we have seen, parametrizing surfaces and determining the orientation induced by a parametrization can be quite complicated. Which aspects of parametric surfaces/orientation of surfaces did you find most challenging? What did you do to address these challenges? • We have also seen that the process of calculating surface integrals (scalar and flux) is often a long process with several steps. For you, what has been the most challenging part about our study of surface integrals? What have you done to address these challenges? • Now that we have seen Stokes' Theorem, we have seen all of the theorems/strategies in this course that help us calculate vector line integrals (for vector fields on R? and R³). Which strategy is your favourite? Which is your least favourite? Are there any that you will need to review more than others before the final exam? • Suppose you have a surface parametrized by r(u, v). In plain language (without mathe- matical notation), explain the geometric significance of the vectors Or dr Or and Or du' dv • In plain language (without mathematical notation), briefly explain what is meant by an orientation of a surface. • If S is an oriented surface with boundary curve C, how do you determine the positive orientation of C? • In plain language (without mathematical notation), explain how Stokes' Theorem can help you calculate a line integral of a vector field on R3. • Suppose F is a vector field on R and you are asked to calculate a line integral F. dr, but it seems too complicated to calculate directly. In a couple of sentences, explain what options you have for strategies to calculate the line integral, and how you would go about determining which strategy to use. • As we have seen, parametrizing surfaces and determining the orientation induced by a parametrization can be quite complicated. Which aspects of parametric surfaces/orientation of surfaces did you find most challenging? What did you do to address these challenges? • We have also seen that the process of calculating surface integrals (scalar and flux) is often a long process with several steps. For you, what has been the most challenging part about our study of surface integrals? What have you done to address these challenges? • Now that we have seen Stokes' Theorem, we have seen all of the theorems/strategies in this course that help us calculate vector line integrals (for vector fields on R? and R³). Which strategy is your favourite? Which is your least favourite? Are there any that you will need to review more than others before the final exam? • Suppose you have a surface parametrized by r(u, v). In plain language (without mathe- matical notation), explain the geometric significance of the vectors Or dr Or and Or du' dv • In plain language (without mathematical notation), briefly explain what is meant by an orientation of a surface. • If S is an oriented surface with boundary curve C, how do you determine the positive orientation of C? • In plain language (without mathematical notation), explain how Stokes' Theorem can help you calculate a line integral of a vector field on R3. • Suppose F is a vector field on R and you are asked to calculate a line integral F. dr, but it seems too complicated to calculate directly. In a couple of sentences, explain what options you have for strategies to calculate the line integral, and how you would go about determining which strategy to use.

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