MTH102 - Multi Variable Calculus
This is a course on Multi-variable calculus. I will be teaching this course jointly with Dr. Anup Biswas. I will be teaching this course for the first two months of Spring 2015 (till the mid-term examination). Dr. Biswas will teach the course after the mid-semester examination. You will find relevant information about this course in this site.
Main reference for this course is
Susan Jean Colley-Vector Calculus.
Other references for this course:-
1. Walter Rudin, Principles of Mathematical Analysis.
2. Calculus- Thomas Finney.
3. Calculus- James Steward.
Please send me an e-mail at debargha@iiserpune.ac.in if you wish to procure the scanned copy of my lecture notes.
Lecture 1(06/01/2015):-
In this lecture, we defined vectors. These are the elements of R^n for n greater than 1. There are four basic concepts in calculus.
1. Limit .
2. Continuity.
3. Differentiation.
4. Integration (inverse operation of differentiation).
We identified the properties of R^n which will help us to study these concepts.
1. R^n is a vector space.
2. R^n is a metric space. This will help us to understand the notion of distance and hence the concept of what is "near" and what is "far".
3. R^n is an inner product space.
We wrote down the properties of inner products. Most importantly, we proved Cauchy-Schwarz inequality and as a consequence triangle inequality.
Lecture 2 (07/01/2015).
This week, we will understand the notion of "distance". Namely, what is the meaning of "far" in the kingdom of vectors.
1. We studied metric space. We defined and gave different examples of metric spaces. We defined open unit ball for arbitrary metric space and as a consequence we defined open unit ball with respect to the most important metric. In particular, we understood how the open unit ball are different
for different metrics.
2. We defined the bounded sequence and convergent sequence using metric space structure of R^n. This is a consequence of whatever we learnt
in MTH 101. Metric space structure will help us to redo the definition of the above concepts from scalar valued functions on set of scalars.
3. We defined convex and path connected sets. We understood the difference of scalars and vectors.
08/01/2015.
Assignment 1 uploaded.
Lecture 3 (13/01/2015):-
In this lecture, we studied the notion of limits. In particular, we understood the concept
Lim_{x ---> a} f(x)=L
for a vector valued function f of vector variables.
Lecture 4 (14/01/2015).
1. We proved that if the limit exist then it is unique.
2. We defined continuous functions. We gave several examples of continuous functions. We also gave several examples of functions, which are not continuous.
3. We prove that a vector valued functions are continuous if and only if the corresponding scalar valued functions are continuous.
We gave several applications of these theorems including that the functions defined by linear transformations are
continuous.
4. We define and give examples of continuous functions.
18/01/2015:-
Assignment 2 Uploaded. Due date 23/01/2015.
Lecture 5 (20/01/2015)
1. Standard Basis vectors and examples of the theorem that vector valued function is continuous if and only if the
corresponding scalar valued functions are continuous.
2. Linear transformations. Examples of linear transformations. What are all linear transformations?
They are continuous.
3. Open and closed subsets, examples, complements of open subsets are closed.
4. If you wish to show some function is continuous it is enough to show inverse image of open subsets are open.
Lecture 6 (21/01/2015).
1. Completed the proof of the above theorem
2. Level curves, gave examples (paraboloid)
3. Path, velocity, speed, acceleration, tangent vector. When can you say a path is smooth?
geometric interpretation of derivatives, different examples of paths, helix.
4. Recall the definition of derivatives from MTH 101, partial functions, equation of tangent plane, existence of partial derivative may not be a guarantee to continuity. Example.
Assignment 3 uploaded (23/01/2015).
Lecture 7 (27/01/2015)
1. Partial derivatives:-
We define partial derivatives for functions R^2 ------> R. Visualize Partial derivatives as graphs in R^3. Computed some examples of partial derivatives.
2. Tangent plane:-
We wrote down the equation of the tangent planes if they exist. We also assert that tangent plane will exist if the partial
derivatives are continuous. Tangent planes should contain tangent lines.
3. Mixed partial derivatives. Give some examples when the mixed partial derivatives may exist but there will not be
any tangent plane. We proved that if the partial derivatives are continuous then mixed second order partial derivatives
agree using Mean Value theorem.
Lecture 8 (28/01/2015).
1. Define total derivatives. To understand total derivatives, we need to understand two concept.
1. Approximation (what is the meaning of english word "approximate"?
2. Linear transformation (English word "line" means they are easier to understand).
A function is said to be differentiable if we can approximate the function near a point by
some simple mapping called linear transformation.
3. In our world, derivative of a function is a matrix. If the function R^n ------> R, then
it will be a matrix in M_12(\R). Called gradient.
4. In general, it will a matrix that you can construct from each scalar valued matrix.
This matrix is called Jacobian matrix.
29/01/2015:
Assignment 4 given.
Please meet your tutors regularly and get your doubts cleared.
1. Tutors and their office Hours:-