1.1.DEFENITIONS
In the study of elementary phisics, several kinds of quatities are encountered, in particular, we distinguish between vectors and scalars.
A scalar is a quantity that is completely characterized by it’s magnitude.
Examples of scalars are numerous: mas, time,volume,etc. A simple extension of the idea of scalar is the scalar field—a function of position that is completely specified by it’s magnitude at all points in space.
A vector is a quantity that is completely characterized by it’s magnitude and direction
1.2.VECTOR ALGEBRA
The sum of two vectors is defined as the vector whose components are the sums of the corresponding components of the original vectors. Thus if C is the sun of A and B, we write:
C = A + B
And
CX = AX + BX, CY = AY + BY, CZ = AZ + BZ
This definitions of the vector sum is completely equivalent to the familiar parallelogram rule for vector addition.
The operation of subtraction is then defined as the addition of the negative and is written:
A – B = A + (-B)
If two vectors are to be multiplied, there are two possibilities, known as the vector and scalar products. Considering first the scalar product, we note that this name derives from the scalar nature of the product, although the alternative names, inner product and dot product, are sometimes used. The definition of the scalar product, written A.B, is
A . B = AXBX + AYBY + AZBZ
This defenition is equivalent to another, and perhaps more familiar, defenition – that is, as the product of the magnitudes of the original vectors times the cosine the angle between these vectors. If A and B are perpendicular to each other,
A . B = 0
The scalar product is commutative. The length of A is
│A│=
It is important to note that cross product depends as on the order of the factors, interchanging the order introduces a minus sign :
B x A = - A x B
Consequently
A x A = 0
The vector product may be easily remembered in terms of a determinant. If i,j, and k are unit vectors – that is, vectors of unit magnitude, in the x-,y-, and z- directions, respectively – then
A x B =
If this determinant is evaluated by the usual rules, the result is precisely our defenition of the cross product.
The preceding algenbraic operations may be combined in many ways. Most of the results so obtained are obvious, however, there are two triple products of sufficient importance to merit explicit mention. The triple scalar product D = A . B x C is easily found to be given by the determinant
D = A . B x C = = - B . A x C
1.3.GRADIENT
The extentions of the ideas introduced in the previous section to differentiation and integration,i.e.,vector calculus, will now be considered. The simplest of these the relation of a particular vector field to the derivatives of a scalar field. It is convenient first to introduce the idea of the directional derivative of a function of several variables, which is just the rate of change of the function in a specified direction. The directional derivative of scalar of a scalar function φ is usually denoted by dφ / ds , it must be understood that ds represents an infinitesimal displacement in the directionbeing considered, and that ds is the scalar magnitude of ds. If ds has the components dx, dy, dz, then
=
=
In order to clarify the idea of a directional derivative, consider a scalar function of two variables. Thus, (x,y) = x2 + y2. The directional derivative at the point x0, y0 depends on the direction. If we choose the direction corresponding to dy/dx = -x0/y0, then we find
│x0,y0 = = = 0
Alternatively, if we choose dy/dx = y0/x0, we find
│xo,yo = = 2
Since ds = . as a third possibility, choose dy/dx = α , then
│xo,yo = (2x0 + 2αy0) (1 + α2)-12
“the gradient of a scalar function is a vector whose magnitude is the maximum directional derivative at the point being considered and whose direction is the direction of the maximum directional derivative at the point”
Equating coefficients of differential of independent variables on both sides of the equation gives
grad = i j k
1.4.VECTOR INTEGRATION
If F is a vector field, a line integral of F is written
Where C is the curve along which the integration is performed, a and b the initial and final points on the curve, and dI an infinitesimal vector displacement along the curve C. Since F.dI is a scalar it is clear that the line integral scalar. The definition of the line integral follows closely the Riemann defenition of the definite integral. The segment of C between a and b is devided into a large number of small increments I, for each increment an interior point is chosen and the value of F at that point found. The scalar product of each increment with the corresponding value of F is found and the sum of these computed. The line integral is then defined as the limit of this sum as the number of increments becomes infinite in such a way that each increment goes to zero. This definition may be compactly written as
=
It is important to note that the line integral usually depends not only on the endpoints a and b but also on the curve C along which the integration is to be done, since the magnitude and direction of F(r) and the direction of dI depend on C and its tangent, respectively. The line integral around a closed curve is of sufficient importance that a special notation is used for it, namely.
If F is again a vector, a surface integral of F is written
Where S is the surface over which the integration is to be performed, da is an infinitesimal are a on S, and n is a unit normal to da. There is a twofold ambiguity in the choice of n, which is resolved by taking n to be the outward drawn normal if S is a closed surface. If S is not closed and is finite, then it has boundary, and the sense of the normal is important only with respect to the arbitarypositive sense of traversing the boundary. The positive sense of the normal is the direction in which a right – hand screw would advance if rotated in the direction of the positive sense on the bounding curve. The surface F over a closed surface S is sometimes denoted by
1.5.DIVERGENCE
Another important operator, which is essentially a derivative, is the divergence operator. The divergence of vektor F, written div F, is defined as follows
The divergence of a vector is the limit of it’s surface integral per unit volume as the volume enclosed by the surface goes to zero. That is
div F =
An extremely important theorem involving the divergence may now be stated and proved
Divergence theorem. The integral of the divergence of a vector over a volume V is equal to the surface integral of the normal component of the vector over the surface bounding V. That is
Consider the volume to be subdivided into a large number of small cells. Let the ith cell have volume V, and be bounded by the surface Si. It’s clear that
1.6.CURL
The curl ineteresting vector differentiall operator is the curl. The curl of a vector, written curl F, is defined as follows.
The curl of a vector is the limit of the ratio of the integral of it’s cross product with the outward drawn normal, over a closed surface, to the volume enclosed by the surface as the volume goes to zero. That is
curl F =
The form of the curl in rectangular coordinates can be easily remembered if it is noted that it is just the expansion of the three-by three determinant, namely,
curl F =
Finding the form of the curl in other coordinate systems is only slightly more complicated and is left to the problem section.
Stoke’s theorem. The line integral of a vector around a closed curve is equal to the integral of the normal component of it’s curl over any surface bounded by the curve. That is
where C is a closed curve that bounds the surface S.
1.7.THE VECTOR DIFFERENTIAL OPERATOR
We now introduce an alternative notation for the three types a vector differentiation that have been discussed-namely, gradient, divergence, and curl. This notation uses the vector differential operator del, defined in Cartesian coordinates as
Del is a differential operator in that it used only in front of a function of(x,y,z), which it differential, it is a vector in that it obeys the laws of vector algebra. In terms of del
grad =
div =
curl =
1.8.FURTHER DEVELOPMENTS
The first of these is the Laplacian operator, which is defined as the divergence of the gradient of a scalar field, and which is usually written 2,
2
In rectangular coordinates,
The curl of the gradient of any scalar field is zero. This statement is most easily verified by writing it out in rectangular coordinates. If the scalar field is , then
Which verifies the original statement. In operator notation,
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Pagi mas alpan. Template baru nih. Sangat menarik dan sangat bagus jika diupdate terus. Pastinya akan banyak visitornya kemari.
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