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VECTOR FUNCTIONS OF A SINGLE VARIABLE
SCALAR AND VECTOR FIELDS
LINE AND SURFACE INTEGRALS
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algebra angle arc length axis basis calculation cartesian coordinates Christoffel symbols component of F compute constant contravariant contravariant components coordinate system covariant components curl F curvilinear coordinates cylindrical coordinates defined definition denote determinant differential directed line segment div F divergence theorem domain dx dy dz e x e equal equations Example Exercise expression field F Find flow lines fluid formula function geometrical given grad f gradient hence identity laplacian Let F line integral magnitude matrix metric tensor nonzero vector normal component notation obtain one-form oriented parallel parallelepiped parametric particle point in space point x,y,z position vector quaternions region represented rotation scalar field scalar multiple Show simply-connected smooth arc Solution sphere surface integral tangent tensor of rank triple scalar product two-form unit vector vector analysis vector field vector product volume integral write xy plane zero-form