* Gradient * Divergence * Curl * Line integrals * Stokes theorem. PPT No. 5

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1 * Gradient * Divergence * Curl * Line integrals * Stokes theorem PPT No. 5

2 Gradient The gradient is a Vector operator It is denoted as It is called as Del or Nabla It operates on a scalar function f to produce a vector, also called as Gradient is a vector differential operator.

3 Gradient Magnitude of the Gradient Its magnitude is the maximum space rate of change of the function at the point of the gradient. Direction of the Gradient It points in the direction of that maximum space rate of change and known as the directional derivative in that direction

4 Gradient is a scalar function, then the gradient of f is the vector whose components are the partial derivatives of f i.e.

5 Gradient Gradient, a vector operator is applied to a real (scalar) function of three variables f (x,y,z). The vector differential operator is given by

6 Gradient In rectangular coordinates with i, j, k as unit vectors in the x, y, z directions In Cylindrical polar coordinates (r, θ, z): In Spherical polar coordinates (r, θ, φ):

7 Divergence The scalar product of Gradient, the vector operator and a vector function A gives a scalar. It is called as Divergence of A and denoted as div A or.a

8 Divergence Divergence is a Scalar Operator that measures how much the vector spreads out (diverges) from the point under consideration. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume element around a given point. A point of positive divergence is a source or faucet (outward flow) A point of negative divergence is a sink or drain (inward flow).

9 Divergence In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. It is a local measure of its spread or outgoing extent. It gives the net flow of the vector field as a result of the exiting and entering flows.

10 Divergence If A is a vector, then its Divergence In rectangular coordinates (x, y, z): In Cylindrical polar coordinates (r, θ, z): In Spherical polar coordinates (r, θ, φ):

11 Curl The cross (vector) product of the vector operator del (Gradient) and a vector function F gives a vector. It is called as curl of F and denoted as curl F or The curl (or rotor) is a vector operator that describes how much the vector field curls (or rotates) around the point under consideration

12 Curl The curl of avector field is defined as the vector field having magnitude equal to the maximum "circulation" at each point and orientation perpendicular to this plane of circulation for each point. The direction of the curl is the axis of rotation, as determined by the Right Hand Rule (If the fingers of right hand are curled in the direction of the vector field; then thumb will point in the direction of curl F).

13 Curl In rectangular coordinates i, j, k are unit vectors in the x, y, z directions then Curl of electrical field E is given by

14 Curl The Curl E expressed in determinant forms as follows: In rectangular coordinates determinant form is The first line is made up of unit vectors, the second line of scalar operators, and the third line of scalar functions

15 Curl In Cylindrical Polar Coordinates (r, θ, z), determinant form of Curl

16 Curl In Spherical Polar Coordinates (r, θ, φ), the determinant form of Curl is:

17 Curl Physically, the curl of a vector field signifies the amount of "rotation" or angular momentum of the contents of given region of space. In the theory of electromagnetism It arises in two out of the four Maxwell equations

18 Line integrals Line integral is (sometimes called a path integral or curve integral It is an integral where the function to be integrated is evaluated along a curve

19 Line integrals If a function f is defined on a curve, the curve is broken up into tiny line segments ds, multiply ds by the function value on the segment and add up all the products. Then a limit is taken as the length of the line segments approaches zero. This new quantity is called the line integral and can be defined in two, three, or higher dimensions

20 Line integrals Consider a function of two variables x, y the values of x and y are the points x, y that lie on a curve C. Let f be a function defined on a curve C of finite length. Then the line integrals of f along C for two & three dimensions are respectively

21 In qualitative terms, a line integral can be thought of as a measure of the total effect of a given field along a given curve. Line integrals More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve

22 Line integrals The line integral can be visualized as the surface created by z = f(x,y) and a curve C in the x-y plane. The line integral of f would be the area of the "curtain" created when the points of the surface that are directly over C are carved out

23 Application of the Line Integral The main application of line integrals is finding the work done on an object in a force field If an object is moving along a curve C through a (electrostatic) force field F, the total work done by the force field is calculated by cutting the curve up into tiny pieces ds The work done W along each piece ds will be approximately equal to dw = F.Tds

24 Application of the Line Integral The work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.

25 Divergence Theorem Let V be the volume of a solid region bounded by a closed surface S oriented with outward pointing unit normal vector n, F be a differentiable vector field (components have continuous partial derivatives). Then

26 Divergence Theorem The volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary. The vector sum of all sources and sinks is equal to the net flow out of a region

27 Divergence Theorem The theorem relates the flow (that is, flux ) of a Vector field through a surface to the behavior of the vector field inside the surface LHS is a volume integral over the volume V, RHS is the surface integral over the boundary of volume V. LHS represents the total of the sources in the volume V, and RHS represents the total flow across the boundary V.

28 Divergence Theorem The divergence theorem provides with a relationship between a triple integral over a solid and the surface integral over the surface that encloses the solid. The divergence theorem is a Conservation Law

29 Divergence Theorem The divergence theorem is an important result in Electrostatics and fluid dynamics It is a special case of the more general Stokes Theorem, which generalizes the Fundamental theorem of calculus