Study notes on Multi Variable Calculus For Electrical Engineering Students

By Yash Bansal|Updated : August 22nd, 2018

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

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

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

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  •  

    Addition of vectors Scalar Multiplication

Dot Productand 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

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An equivalent definition of the dot product is

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

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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

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  • Example

    For the function

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    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:

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

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    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:

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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)=4x^2+y^2. The gradient is

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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

displaymath94Here is an example. Let

displaymath96The divergence is given by:

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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

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 An alternative notation is


displaymath102The above formula for the curl is difficult to remember. An alternative formula for the curl is

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 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

 displaymath106As you can see, this term is part of the x-component of the curl.

ExampleF=<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 thatdisplaymath42

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 statesdisplaymath57

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:

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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.

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