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Notes on Conics, Spheres, Cylinders and Vectors
Unit I: Conics and Their Properties
- General equation of the second degree
- Classification of conic sections
- Centre, asymptotes, axes, eccentricity, foci, and directrices of conics
- Tangent at any point to a conic
- Chord of contact
- Pole of a line to a conic
- Director circle of a conic
- Polar equation of a conic
- Tangent and normal to a conic
- Confocal conics
Unit II: Spheres and Cones
- Sphere:
- General form
- Plane section of a sphere
- Sphere through a given circle
- Intersection of two spheres
- Tangent plane and line
- Polar plane and line
- Orthogonal spheres
- Radical plane of two spheres
- Co-axial system of spheres
- Cone:
- Equation of a cone
- Right circular cone
- Quadric cone
- Enveloping cone
- Tangent plane and condition of tangency
Unit III: Cylinders and Central Conicoids
- Cylinder:
- Right circular cylinder
- Enveloping cylinder
- Central Conicoids:
- Equation of tangent plane
- Director sphere
- Normal to the conicoids
- Polar plane of a point
- Enveloping cone of a conicoid
- Enveloping cylinder of a conicoid
- Confocal conicoid
- Reduction of second degree equations
Unit IV: Vector Calculus
- Scalar and vector product of three vectors and four vectors
- Reciprocal vectors
- Vector differentiation and derivative along a curve
- Directional derivatives
- Gradient of a scalar point function
- Divergence and curl of vector point functions
- Geometrical meanings and vector identities
- Vector integration:
- Line integral
- Surface integral
- Volume integral
- Theorems:
- Gauss Theorem
- Green’s Theorem
- Stoke’s Theorem
- Problems based on these theorems
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Notes on Conics, Spheres, Cylinders and Vectors
Unit I: Conics
- General equation of second degree
- Classification of conic sections
- Centre, asymptotes, axes, eccentricity, foci, and directrices of conics
- Tangent at any point to a conic
- Chord of contact
- Pole of line to a conic
- Director circle of a conic
- Polar equation of a conic
- Tangent and normal to a conic
- Confocal conics
Unit II: Spheres and Cones
- Sphere:
- General form
- Plane section of a sphere
- Sphere through a given circle
- Intersection of two spheres
- Tangent plane and line
- Polar plane and line
- Orthogonal spheres
- Radical plane of two spheres
- Co-axial system of spheres
- Cone:
- Equation of a cone
- Right circular cone
- Quadric cone
- Enveloping cone
- Tangent plane and condition of tangency
Unit III: Cylinders and Central Conicoids
- Cylinder:
- Right circular cylinder
- Enveloping cylinder
- Central Conicoids:
- Equation of tangent plane
- Director sphere
- Normal to the conicoids
- Polar plane of a point
- Enveloping cone of a conicoid
- Enveloping cylinder of a conicoid
- Confocal conicoid
- Reduction of second degree equations
Unit IV: Vector Calculus
- Scalar and vector product of three and four vectors
- Reciprocal vectors
- Vector differentiation and derivative along a curve
- Directional derivatives
- Gradient of a scalar point function
- Divergence and curl of vector point functions and their geometrical meanings
- Vector identities
- Vector integration:
- Line integral
- Surface integral
- Volume integral
- Theorems:
- Gauss Theorem
- Green’s Theorem
- Stoke’s Theorem
- Problems based on these theorems
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