Infinitesimal Calculus II

Code:

M112

Acronym:

M112

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 2S

Active?	Yes
Web Page:	http://elearning2.fc.up.pt/aulasweb0910/course/view.php?id=1300
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	31	Plano de Estudos a partir de 2008	1	-	7,5	-
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:CC	1	Plano de estudos de 2008 até 2013/14	2	-	7,5	-
			3
L:F	45	Plano de estudos a partir de 2008	1	-	7,5	-
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-
L:M	102	Plano de estudos a partir de 2009	1	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-
MI:EF	54	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

To know how to identify the graphs of quadratic equations in two and three real variables.

To know the basic concepts about calculus of parametrized curves in the plane and the space.

To know the fundamental results concerning the analysis of multivariate functions aiming the understanding of the concepts of partial derivative, gradient vector, local maxima and minima, tangent plane to the graph of functions of two variables, and the method for finding extreme values of constrained functions.

To know the methods of multiple integration and use them to determine areas, volumes, etc, of bounded plane or space regions, using change of variables if necessary.

Program

1. Conic sections and quadratic surfaces in the space.
The euclidean vector space R^n. Diagonalization of quadratic forms. Conic sections and quadratic surfaces in the space.

2. Parametrized curves.
Velocity, acceleration, curvature, Frenet frame.

3. Differential calculus of vector-valued multivariate functions.
Graphs of real-valued functions of two variables, contour lines of functions of two variables and level surfaces of functions of three variables. Open and closed subsets of R^n. Accumulation point and isolated point. Limits and continuity of functions. Directional derivatives and partial derivatives. Derivative function at a point of a real-valued multivariate function. Gradient vector and derivability. Tangent plane to the graph of a function of two variables. Interpretation of the gradient vector. Normal line and tangent hiperplane at a point on the level surface of a function. Higher order derivatives. Derivative function at a point of a vector-valued multivariate function. Jacobian matrix. Derivation of composition of functions. Examples. Inverse function theorem. Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes. The method of Lagrange multipliers for finding extreme values of constrained functions.

4. Multiple integrals.
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem. Calculation of double and triple integrals via iterated integrals. Integration and the change of coordinates. Applications: double integrals in polar coordinates, and triple integrals in cylindrical and spherical coordinates.

Mandatory literature

000046967. ISBN: 0-387-96405-3
000097905. ISBN: 978-0-495-38273-7

Complementary Bibliography

J.Marsden and A.Weinstein; Calculus. Vol. III. , New York: Springer-Verlag, 1985
000043894. ISBN: 0-12-232550-8
R.Larson, R.P.Hostetler and B.H.Edwards; Calculus (8th Edition), Houghton Mifflin Company , 2006

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	70,00
	Total:	-	0,00

Eligibility for exams

Exam or four tests during the semester. Students with final mark over seventeen (out of twenty) will have an extra test.

Calculation formula of final grade

Exam or four tests during the semester. Students with final mark over seventeen (out of twenty) will have an extra test.