## Calculus 2

 Faculteit Science and Engineering Jaar 2020/21 Vakcode WBMA029-05 Vaknaam Calculus 2 Niveau(s) propedeuse Voertaal Engels Periode semester II a ECTS 5 Rooster rooster.rug.nl

Uitgebreide vaknaam Calculus 2
Leerdoelen 1. The student is able to compute the lengths of parametric curves, reparametrize curves by arclength and compute the scalar curvature of a curve.
2. The student knows the definition of continuity for vector valued and multivariable functions, and is able to check continuity for such functions.
3. The student 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. The student is able to apply the chain rule.
4. The student is able to 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. The student knows the Implicit Function Theorem and is able to apply it.
6. The student is able to compute double and triple integrals in Cartesian, polar, spherical and cylindrical coordinates.
7. The student is able to compute line integrals of scalar functions and vector fields.
8. The student is able to 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. The student knows and is able to apply the integral theorems of Green, Stokes, and Gauss.
Omschrijving 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 course 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.
Uren per week
Onderwijsvorm Hoorcollege (LC), Werkcollege (T)
Toetsvorm Opdracht (AST), Schriftelijk tentamen (WE)
(Homework assignments, midterm exam and final exam. If the mark for the final exam is 4.5 or higher then the grade for the course will be determined by max(FE, 0.2 HW + 0.2 ME + 0.6 FE) where FE is the mark for the final exam, HW is the average mark for the homework assignments and ME is the mark for the midterm exam. If the mark for the final exam lower than 4.5 then the grade for the course will be the mark for the final exam. If a resit exam is taken then the grade for the course will be the mark of the resit exam (i.e. homework and the midterm exam are not taken into account in this case).)
Vaksoort propedeuse
Coördinator prof. dr. H. Waalkens
Docent(en) prof. dr. H. Waalkens
Verplichte literatuur
Titel Auteur ISBN Prijs
Calculus, Early Transcendentals, International Metric Edition (newest edition) James Stewart 9781305272378
Entreevoorwaarden The course unit assumes prior knowledge acquired from Calculus 1 and Linear Algebra 1.
Opmerkingen The concepts developed in Calculus 2 are used in many advanced courses in mathematics and physics. They are in particular a
prerequisite for the course Analysis on Manifolds where the concepts are generalized with the help of differential forms.

This course was registered last year with course code WPMA13001
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