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VECTOR FUNCTIONS OF A SINGLE VARIABLE
SCALAR AND VECTOR FIELDS
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A X B algebra angle arc length axes axis basis Cartesian coordinates Christoffel symbols component of F compute constant contravariant contravariant components coordinate system cosine covariant components curl F curvilinear coordinates defined definition denote density determinant differential forms directed line segment div F divergence theorem domain equal equations example Exercise expression field F Figure Find flow lines fluid formula function geometrical given gives grad gradient hence identically Laplacian Let F line integral magnitude matrix metric tensor nonzero vector normal component notation obtain one-form oriented parallel parallelepiped parameter particle perpendicular point in space point x,y,z position vector quaternions reader region represented rotation scalar field scalar multiple Section 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 vector zero-form