## Vector Calculus |

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### Contents

CHAPTER | 1 |

Vector function | 2 |

Derivative of a vector function with respect to a scalar | 3 |

Curves in space 10 | 10 |

Velocity and acceleration | 12 |

Integration of vector functions CHAPTER 2 | 22 |

Gradient Divergence and Curl 3074 1 Partial derivatives of vectors 30 2 The vector differential operator Del V 30 | 30 |

Gradient of a scalar field | 31 |

Divergence of a vector point function | 49 |

The Laplacian operator V2 51 | 51 |

Important vector identities 72 | 56 |

Greens Gausss and Stokes Theorems 75168 1 Some preliminary concepts | 75 |

Circulation | 78 |

Volume integrals | 80 |

Greens theorem in the plane 96 | 96 |

The divergence theorem of Gauss | 105 |

Level Surfaces | 37 |

Directional derivative of a scalar point function | 38 |

Tangent plane and normal to a level surface 41 | 41 |

Stokes theorem 1 152 | 132 |

Line integrals independent of path | 152 |

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### Common terms and phrases

added any constant Agra Allahabad Bombay boundary circle closed surface component cos2 curl F curl F-n dS curl F=0 curl grad defined denote directional derivative div F divergence theorem dr dr dt dt dx dy dz exact differential F»dr field F function f given line integral grad f Green's theorem Hence independent of path irrotational Kanpur Kerala Let F level surface Meerut outward drawn normal parametric equations partial derivatives particle perpendicular piecewise smooth position vector Proof Prove that curl Prove that div Rohilkhand scalar field scalar function scalar point function scalar quantity scalar variable simply connected sin2 Solution SOLVED EXAMPLES Ex sphere straight line Suppose surface integral tangent plane theorem in plane unit normal vector unit vector vector field vector function vector normal vector point function velocity Verify Stoke's theorem volume enclosed volume integral whence xy-plane zero