VECTOR CALCULUS
MAXWELL'S EQUATIONS
POINT FORM
INTEGRAL FORM
BASICS CONCEPTS IN VECTOR CALCULUS IN CARTESIAN COORDINATE SYSTEMS
DIFFERENTIAL DISPLACEMENT
DIFFERENTIAL NORMAL AREA
DIFFERENTIAL VOLUME
BASICS CONCEPTS IN VECTOR CALCULUS IN CYLINDRICAL COORDINATE SYSTEMS
DIFFERENTIAL DISPLACEMENT
DIFFERENTIAL NORMAL AREA
DIFFERENTIAL VOLUME
THE AREA ELEMENTS IN CYLINDRICAL SYSTEMS
BASICS CONCEPTS IN VECTOR CALCULUS IN SPHERICAL COORDINATE SYSTEMS
THE DIFFERENTIAL DISPLACEMENT
THE DIFFERENTIAL NORMAL AREA
THE DIFFERENTIAL VOLUME
THE AREA ELEMENTS IN SPHERICAL SYSTEMS
LINE,SURFACE AND VOLUME INTEGRALS
Del OPERATOR
ALSO KNOWN AS THE GRADIENT OPERATOR
IS NOT A VECTOR IN ITSELF,BUT WHEN IT OPERATES ON A SCALAR FUNCTION,THE PRODUCT IS A VECTOR
USEFUL IN FINDING THE GRADIENT OF A scalar V
USEFUL IN FINDING THE DIVERGENCE OF A vector A
USEFUL IN FINDING THE CURL OF A vector A
USEFUL IN FINDING THE Laplacian OF A scalar V
In CYLINDRICAL COORDINATE SYSTEM
In SPHERICAL COORDINATE SYSTEM
GRADIENT OF A SCALAR
THE GRADIENT OF A SCALAR field V is A VECTOR THAT REPRESENTS BOTH THE MAGNITUDE AND THE DIRECTION OF THE MAXIMUM SPACE RATE OF INCREASE of V
CARTESIAN COORDINATES
CYLINDRICAL COORDINATES
SPHERICAL COORDINATES
DIVERGENCE OF A VECTOR AND DIVERGENCE THEOREM
IS THE OUTWARD FLUX PER UNIT VOLUME AS THE VOLUME SHRINKS about P.
CARTESIAN SYSTEM
CYLINDRICAL SYSTEM
SPHERICAL SYSTEM
DIVERGENCE THEOREM
THE TOTAL OUTWARD FLUX OF A VECTOR field A through a CLOSED surface S is THE SAME AS THE VOLUME INTEGRAL OF THE DIVERGENCE of A.
NOTE : A CLOSE SURFACE INSIDE A VECTOR FIELD HOLDS NO NET FLUX; THE TOTAL FLUX INTO THE SURFACE WILL GO OUT OF THE SURFACE COMPLETELY!
CURL OF VECTOR
THE CURL of A IS AN AXIAL (OR ROTATIONAL) VECTOR WHOSE MAGNITUDE IS THE MAXIMUM CIRUCLATION of A per unit AREA AS THE AREA LENDS TO ZERO AND WHOSE DIRECTION IS THE NORMAL DIRECTION OF THE AREA WHEN THE AREA IS ORIENTED SO AS TO MAKE THE CIRCULATION MAXIMUM
CURL IN DETERMINANT FORM
CARTESIAN SYSTEM
CYLINDRICAL SYSTEM
SPHERICAL SYSTEM
STROKE'S THEOREM
STATES THAT THE CIRCULATION OF A VECTOR field A around a (closed) path L is equal to the SURFACE INTEGRAL OF THE CURL of A over the open surface S bounded by L provided that A and .... are continuous on S.
THE PROOF OF STROKE'S THEOREM
LAPLACIAN OF A SCALAR
IN CARTESIAN COORDINATES
IN CYLINDRICAL COORDINATES
IN SPHERICAL COORDINATES
CURL AND DIV FOR CLASSIFICATION OF VECTOR FIELDS
DEFINITIONS: SOLENOIDAL AND POTENTIAL
HELMHOLTZ'S THEOREM
CONSERVATIVE VECTOR FIELD
A vector field A is conservative if its line integral possesses of these two properties :-
THE LINE INTEGRAL FROM A POINT P TO A POINT Q IS INDEPENDENT OF THE PATH
THE LINE INTEHRAL AROUND ANY CLOSED PATH IS ZERO