Engineering Mathematics : Differential Calculus & Vector Calculus

Limits

Let us consider a function f(x) defined in an interval l. if we see the behavior of f(x) become closer and closer to a number l as x  a then l is said to be limit of f(x)at x=a.

Left-Hand Limit –

Let  function f(x) is said to approach l as x →a from left if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that 

Right Hand Limit

Let  function f(x) is said to approach l as x → a from right if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that

L- Hospital Rule:

When 

must not be zero, where f⁽ⁿ⁾ and g⁽ⁿ⁾ are n^th derivative of f(x) and g(x).

L- Hospital Rule for the form (∞ – ∞, 0 × ∞):

L-Hospital Rule for the form (0°, 1^∞, ∞°):

CONTINUITY

A function y = f(x) is said to be continuous if the graph of the function is a continuous curve. On the other hand, if a curve is broken at some point say x = a, we say that the function is not continuous or discontinuous.

Definition:

A function f(x) is said to be continuous at x = a if and only if the following three conditions are satisfied:

Properties of continuous functions:

(i) A function which is continuous in a closed interval is also bounded in that interval.

(ii) A continuous function which has opposite signs at two points vanishes at least once between these points and vanishing point is called root of the function.

(iii) A continuous function f(x) in the closed interval [a, b] assumes at least once every value between f(a) and f(b), it being assumed that

 f(a) ≠ f(b).

DIFFERENTIABILITY

Note.

A Necessary condition for the Existence of a Finite Derivative

Continuity is necessary but not Sufficient for the existence of finite derivatives.

Fundamental Theorem:

Rolle’s Theorem:

If

(i)   f(x) is continuous is the closed interval [a, b],

(ii)  f’(x) exists for every value of x in the open interval (a, b) and

(iii) f (a) = f(b), then there is at least one value c of x in (a, b) such that f’ (c) = 0.

Lagrange’s Mean-Value Theorem:

If

 (i) f(x) is continuous in the closed interval [a, b], and

(ii) f’(x) exists in the open interval (a, b),

 then there is at least there is at one value c of x (a, b),

Cauchy’s Mean-value theorem:

If (i) f(x) and g(x) be continuous in [a, b]

(ii) f’(x) and g’(x) exist in (a, b) and

(iii) g’(x) ≠ 0 for any value of x in (a, b),

Basic Differentiation Formulas.

Suppose f  and g are differentiable functions, c is any real number, then

Vectors

• The Scalars are quantities that only have a magnitude like mass,  field strength. Many times it is often useful to have a quantity that has not only a magnitude but also a direction; such a quantity is called a vector. Examples of quantities represented by vectors include velocity, acceleration, and virtually any type of force (frictional, gravitational, electric, magnetic, etc.)

• The magnitude (or length) of a vector v with initial point (x_1,y_1,z_1) and terminal point (x_2,y_2,z_2) is

  

  Vectors obey the natural intuitive laws of addition and scalar multiplication:

  

  The figures below illustrate the operations of addition and scalar multiplication in the two-dimensional case.

   

•  

  Addition of vectors Scalar Multiplication

Dot Product and Cross product:

here dot product of two vectors a=<a_1,a_2,a_3> and b=<b_1,b_2,b_3> is given by

An equivalent definition of the dot product is

The cross product of two vectors a=<a_1,a_2,a_3> and b=<b_1,b_2,b_3> is given by

Functions of Two or More Variables:

1. Partial Derivatives

Differentiating a function of more than one variable is more complicated than differentiating a function of one variable. For a function of several variables, the rate of change of the function depends on direction!. Consider the function

• Example

  For the function

  

  find the partial derivatives of f with respect to x and y and compute the rates of change of the function in the x and y directions at the point (-1,2). Initially we will not specify the values of x and y when we take the derivatives; we will just remember which one we are going to hold constant while taking the derivative. First, hold y fixed and find the partial derivative of f with respect to x:

  

  Second, hold x fixed and find the partial derivative of f with respect to y:

  

  Now, plug in the values x=-1 and y=2 into the equations. We obtain f_x(-1,2)=10 and f_y(-1,2)=28.

2. The Gradient and Directional Derivative:

The gradient of a function w=f(x,y,z) is the vector function:

For a function of two variables z=f(x,y), the gradient is the two-dimensional vector <f_x(x,y),f_y(x,y)>. This definition generalizes in a natural way to functions of more than three variables.

Examples

For the function z=f(x,y)=4x²+y². The gradient is

                                 grad f=<8x, 2y>

3. Divergence and Curl of Vector Fields:

Divergence of a Vector Field

The divergence of a vector field F=<P(x,y,z),Q(x,y,z),R(x,y,z)>, denoted by div F, is the scalar function defined by the dot product

Here is an example. Let

The divergence is given by:

Curl of a Vector Field

The curl of a vector field F=<P(x,y,z),Q(x,y,z),R(x,y,z)>, denoted curlF, is the vector field defined by the cross product

 An alternative notation is

The above formula for the curl is difficult to remember. An alternative formula for the curl is

 Det means the determinant of the 3x3 matrix. Recall that the determinant consists of a bunch of terms which are products of terms from each row. The product of the terms on the diagonal is

As you can see, this term is part of the x-component of the curl.

Example: F=<xyz,ysin z, ycos x>.

curl F = <cos x - ycos z, xy + ysin z, -xz>.

4. Line Integrals:

Green's Theorem:

Green's Theorem states that

Here it is assumed that P and Q have continuous partial derivatives on an open region containing R.

5. Surface Integrals:

Stokes' Theorem: Stokes' Theorem relates line integrals of vector fields to surface integrals of vector fields.

Stokes' Theorem states

In general C is the boundary of S and is assumed to be piecewise smooth. For the above equality to hold the direction of the normal vector n and the direction in which C is traversed must be consistent. Suppose that n points in some direction and consider a person walking on the curve C with their head pointing in the same direction as n. For consistency C must be traversed in such a way so that the surface is always on the left.

The Divergence Theorem 

The Divergence Theorem relates relates volume integrals to surface integrals of vector fields. Let R be a region in xyz space with surface S. Let n denote the unit normal vector to S pointing in the outward direction. Let F(x,y,z)=<P(x,y,z),Q(x,y,z),R(x,y,z)> be a vector field whose components P, Q, and R have continuous partial derivatives. The Divergence Theorem states:

Here div F is the divergence of F. There are various technical restrictions on the region R and the surface S; see the references for the details. The theorem is valid for regions bounded by ellipsoids, spheres, and rectangular boxes.

6. Volume Integrals:

GREEN’S THEOREM

If R is a closed region in the x-y plane bounded by a single closed curve C and if M (x, y) and N (x, y) are a continuous function of x and y having continuous derivative in R then

STROKES THEOREM

 GAUSS DIVERGENCE THEOREM

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