Calculus 2
| Faculty | Science and Engineering |
| Year | 2023-2024 |
| Course code | WBMA029-05 |
| Level(s) | Bachelor |
| Credits (ECTS) | 5 |
| Schedule | VIEW SCHEDULE | ||||||||||
| Period | Semester 2a (2024-02-05 - 2024-04-14) | ||||||||||
| Language of instruction | English | ||||||||||
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| Remarks |
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| Extended course name |
Calculus 2
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| Prerequisites |
The course unit assumes prior knowledge acquired from Calculus 1 and Linear Algebra 1.
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| Learning outcomes |
At the end of the course, the student is able to:
1. compute the lengths of parametric curves, reparametrize curves by arclength and compute the scalar curvature of a curve. 2. knows the definition of continuity for vector valued and multivariable functions, and is able to check continuity for such functions. 3. has a firm understanding of differentiability for vector valued and multivariable functions including partial and directional derivatives and is able to compute partial and directional derivatives from the definition, linear approximations, and determine the tangent plane of the graph of a multivariable function. He/she is able to apply the chain rule. 4. determine extrema of multivariable functions. This also includes extrema under a constraint by applying the method of a Lagrange multiplier. For functions of two variables, the students can use the Hessian to distinguish between minima, maxima and saddle points. 5. knows the Implicit Function Theorem and is able to apply it. 6. compute double and triple integrals in Cartesian, polar, spherical and cylindrical coordinates. 7. compute line and surface integrals of scalar functions and vector fields. 8. compute the curl and the divergence of a vector field and has an idea of the geometric meaning. The student is able to determine whether a vector field is conservative, and compute a potential function for a conservative vector field. 9. knows and is able to apply the integral theorems of Green, Stokes, and Gauss. |
| Description |
The course Calculus 2 gives a classical introduction to the field of multivariable calculus. The course proceeds the course Calculus 1 which mainly concerns the calculus of functions of a single variable, and generalizes many of its concepts like continuity, differentiability and integration to the case of multivariable functions. Apart from the background established in Calculus 1 many aspects from the course Linear Algebra 1 like vectors, linear maps, matrices and inner products are heavily used in Calculus 2. In Calculus 2 the means are developed to compute the work required to displace a body along a curve through a given force field or to compute the amount of fluid flowing through a surface in a period of time from the velocity field of the fluid. The course culminates
in the study of the integral theorems by Green, Stokes and Gauss which form, e.g., the basis for formulating Maxwell’s Equations of electrodynamics. The concepts developed in Calculus 2 are used in many advanced courses in mathematics and physics. They are in particular a prerequisite for the courses Multivariable Analysis and Analysis on Manifolds where the concepts are generalized with the help of differential forms. More concretely, the topics addressed in Calculus 2 are spatial curves together with their parametrization by arclength and their curvature, continuity of vector valued functions and multivariable functions, partial and directional derivatives, the linear approximation of a multivariable function, the chain rule for multivariable functions, the tangent plane of the graph of a multivariable function, extrema of multivariable functions and of multivariable functions with constraints using the method of Lagrange multipliers, multiple integrals, the Jacobian, integration of vector fields along curves and over surfaces, conservative vector fields and potential functions, the curl and divergence of vector fields, and Green’s, Stokes', and Gauss' Theorems. |
| Hours per week | 0 |
| Teaching method | Assignment, Lecture, Tutorial |
| Assessment |
Exam, Written
Interim test
Assignment
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If the mark for the final exam is 4.5 or higher then the grade for the course will be max(FE, 0.2 HW + 0.2 ME + 0.6 FE) where FE, HW and ME are the grades for the final exam, homework assignments and midterm exam, respectively. If the mark for the final exam is lower than 4.5 then the grade for the course will be the mark for the final exam. HW and ME do not count in case of a resit exam.
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