Course

Calculus of vector-valued functions: Vector-valued functions of a real variable Algebraic operations. Components- Limits, derivatives and integrals-Applications to curves. Tangency- Applications to curvilinear motion-Velocity, speed and acceleration-The unit tangent, the principal normal -The definition of arc length. 

Vol.1, Chapter 14- Sec. 14.1 to 14.10. 

Differential calculus of scalar and vector fields: Functions of ����to����.Scalar and vector fields-Open balls and open sets-Limits and continuity-The derivative of a scalar field with respect to a vector-Directional derivatives and partial derivatives Partial derivatives of higher order-Directional derivatives and continuity-The total derivative-The gradient of a scalar field-A chain rule for derivatives of scalar fields Applications to geometry. Level sets. Tangent planes 

Vol.2, Chapter-8-Sec. 8.1 to 8.17.

Line Integrals: Introduction-Paths and line integrals-Other notations for line integrals Basic properties of line integral-Open connected sets. Independence of paths-The second fundamental theorem of calculus for line integrals-The first fundamental theorem of calculus for line integrals-Necessary and sufficient conditions for a vector field to be gradient-Necessary conditions for a vector field to be gradient-Special methods for constructing potential functions. 

Vol.2, Chapter-10-Sec 10.1 to 10.5, 10.10 and 10.11, 10.14 to10.18. 

Multiple Integrals: Introduction-Green’s theorem in the plane-Some applications of Green’s theorem-A necessary and sufficient condition for a two-dimensional vector field to be a gradient-Change of variables in double integral-Special cases of transformation formula. 

Vol.2, Chapter-11-Sec. 11.19 to 11.22, 11.26 to 11.28. 

Surface Integrals: Parametric representation of a surface-The fundamental vector product- The fundamental vector product as a normal to the surface-Surface integrals-Other notations for surface integrals-The theorem of Stokes-The curl and divergence of a vector field- Further properties of the curl and divergence-The divergence theorem (Gauss’ theorem) 

Vol.2, Chapter-12-Sec. 12.1 to 12.4, 12.7,12.9 to12.15, 12.19 and 12.21.

Text Books
Howard Anton, IRl Bivens, Stephens Davis, “Calculus” Wiley, 10th Edition, 2016 Reprint. Tom M. Apostol, CalculusVolume1, John Wiley & Sons, Second edition, 2007. Tom M. Apostol, Calculus Volume 2, John Wiley & Sons, Second edition, 2007.

References

1. Howard Anton “Calculus” John Wiley and Sons
2. Murray R Spiegel, Theory and problems of vector analysis, Schaum’s outline series, McGraw-Hill Book Compnay 1974.
3. Finney and Thomas , Calculus, Pearson, Eleventh Edition, 2008.