MAT Math. Tools II Tangent Plane and Normal Suppose the scalar field  =  ( x, y, z, t) at time t o, the level surfaces are given by  ( x, y, z, t o ) = c’ i.e.  ( x, y, z ) = c P The plane through P, perpendicular to is called the tangent plane.

Equation of the Tangent Plane P roro r Since PQ = r - r o lies on the tangent plane,it is perpendicularto. So the equation of the tangent plane is Q Cartesian Equation Let r o = ( x o, y o, z o ), r = ( x, y, z ) and Above equation becomes

Cartesian Equation of the Normal becomes E.g. For the scalar field φ = ( x - y ) 2 +2t x y + t 2 z, find the equation of the tangent plane and the normal at the point A ( 1, 2, 1 ) when t =1. SolutionWhen t =1 level surfaces are given by ( x - y ) 2 +2 x y + z = c

i.e. x 2 + y 2 + z = c For the level surfaces through A, = c So, c = 6 At A, So, the equation of the tangent plane is

So, the equation of the normal is

Vector fields Field lines Suppose A is a vector field, and l is a curve in the domain of the vector field. At any point P on l, A(P) is parallel to the tangent to l at P, then l is called a field line. Equation of a Field line drdr P Suppose that A = ai + bj + ck, then dr = ( ai + bj + ck ) i.e. dx i + dy j + dz k = ( ai + bj + ck ) So dx = a, dy = b, dz = c. l

We represent the field lines as the curve of intersection of two such surfaces. Solutions represents surfaces. Differential equation of a field line We solve this equation pair wise.

E.g.Find the field lines of A = x 2 yz i + xy 2 z j + xyz 2 k solutionDifferential equation of a field line So, field lines are given by the curve of the intersection of the surfaces

Since x = Ay and y = Bz means equation of planes the intersection is a straight lines. So, field lines are given by E.g.Find the field lines of solutionDifferential equation of a field line So, field lines are given by the curve of the intersection of the surfaces

E.g.Find the field lines of E.g.Find the field lines of

r P O Here OP= r. x = r cos θ, y = r sin θ X Y Metric Systems of coordinates … in 2-D y x Polar coordinates

r P O X Y Unit Vectors

Different types of coordinates in 3-D 1. Cylindrical polar coordinates z Q P O Here OQ= r. So P. x = r cos, y = r sin and z = z. X Y Z Metric

z Q P 2. Spherical polar coordinates Here OP = r. So P. x = r sin cos, y = r sin sin. Z = r cos X Y Z Metric

In 3-dimension with respect to a mutually perpendicular system of coordinates Ouvw, the metric is given by The unit vectors in the increasing directions of u, v and w are, and respectively. Curvilinear Coordinates So the unit vectors are given by, and

For a scalar field the gradient is For the vector field

Values for different system of Coordinates Cylindrical Polar 1r1 Spherical Polar 1rr sin

With Cylindrical polar cdts

With Spherical polar cdts

Small Area in 2-D with Oxy cdts X Y x = c dx y = k dy Area = dx dy

Small Area in 2-D with Polar cdts r = c dr Area =

With Cylindrical Polar Cdts Volume element is given by dz dr Small Volume in 3-D

With Spherical Polar Cdts Volume element is given by Small Area in 3-D