Definition of Line integrals of Scalar and vector fields, and the geometrical interpretation of scalar fields is discussed in this lecture. Some basic notions are also covered in the mid - including fundamental theorem of calculus, tangent vector and smooth curves etc.

Now we will use the definition of line integrals of scalar fields to evaluate them. Two computational examples are worked.

Now we will evaluate the line integrals of vector fields. Two computational examples are worked.

First we will understand how to integrate along a PWS curve. Then we will solve a simple example by using this knowledge.

PDF file also contains the examples of next lecture.

Two more examples of line integrals along PWS curves are solved in this lecture. These examples are little bit more difficult than example-1 of preceding lecture.

Example No. was written wrongly in both of these examples by mistake. 

Line integral of a vector field F around a curve C is the work done by F around C, as illustrated in this lecture. One example is also solved.

Two more examples of calculating work done are worked in this lecture. Degree of difficulty increases by a fraction as compared to previous example i.e. Example#1.

These two examples of calculating the work done are very important to learn. In the first example i.e. Example#4, you will see animation for your understanding, and MATLAB code is used to plot the curve described by parametric equations. You can use this code to plot any curve given parametrically.

The notion of Path independent line integrals is discussed in detail in this lecture. The concept of Conservative Fields is also given. The mathematical conditions under which a field is conservative are also highlighted (proved in next lectures in form of theorems). At the end of lecture, nonconservative fields are discussed shortly.

We will now state and prove Theorem #1 of this chapter. It states that the line integral of F is path independent in a domain D if and only if the line integral of F is zero along every closed path in D.

The proof is very naive.

This is very important theorem which relates Conservative fields with Path independent Line integrals (or work done). Two corollaries of this theorem are also discussed.

The proof is optional-you may leave it! but you should try such technical proofs.  

Now we will learn how to find a potential for a conservative field. Two examples are solved using different methods. These are very important examples - problems of this kind often appear in examination.

The concept of simply connected domains is illustrated with the help of animations. Simple examples are considered in this lecture.

PDF file of next lecture is also included.

3D examples are focused in this lecture. I hope the students may like the work.

Two more theorems regarding path independence of line integrals are stated and proved in this lecture.

The proofs are straightforward, but the converse of Theorem #4 is not included as it requires the Stoke's Theorem (hence it will be proved in chp.03)

Now the students will learn to apply Theorem#2 and Theorem#3 with the help of two worked examples.

In these examples we will learn how to apply these theorems to establish path independence of a given line integral and then evaluating these integrals.

NOTE: In the Example#2 the integral is not evaluated (left as your exercise)

Now the students will learn to apply Theorem#4 with the help of two worked examples.

In these examples we will learn how to apply this theorem to establish path independence of a given line integral and then evaluating these integrals.

Simply connectedness of a domain is sufficient to imply the path independence of line integral of a conservative field. This fact is illustrated with the help of two examples in this lecture.

Now we will state and prove the Law of Conservation of Energy. The proof is very simple and identical to that you may have learned in Mechanics.

Now we will start discussing the applications of line integrals and solve the corresponding examples.

This lecture presents the first application i.e. calculation of Area of a Fence. Two examples are solved in this regard.

Now we will learn to apply Line integrals to calculate the mass of a wire. One example is worked for this.

As the 3rd application, we will learn to determine the center of mass of a wire using line integrals. One example is solved for illustration.

As the last (but not least) application, we will learn to calculate the Moment of inertial of a wire about an axis using line integrals. One example is worked.

This completes chapter 1. I hope the students may be now in position to solve any problem regarding line integrals of scalar and vector fields.