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MAT 128 1.0 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.

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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

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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

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i.e. x 2 + y 2 + z = c For the level surfaces through A, 1 2 + 2 2 + 2 = c So, c = 6 At A, So, the equation of the tangent plane is

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So, the equation of the normal is

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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

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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.

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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

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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

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

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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

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r P O X Y Unit Vectors

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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

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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

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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

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For a scalar field the gradient is For the vector field

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

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With Cylindrical polar cdts

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With Spherical polar cdts

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

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

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

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

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homogeneous coordinates equationmisc point(w ; x, y, z)r w = S 0 where S 0 = xi + yj + zk 3 points in 3D space.

homogeneous coordinates equationmisc point(w ; x, y, z)r w = S 0 where S 0 = xi + yj + zk 3 points in 3D space.

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