Vector Analysis and Cartesian Tensors

a₁ axes Ox₁x2x3 axes Oxyz b₁ b₂ CARTESIAN TENSORS chain rule closed region components constant continuously differentiable coordinate axes coordinate system curl curl curl curl F cylindrical polar coordinates defined definition denote direction cosines div F divergence theorem double integral dx dy eigenvalues Evaluate example EXERCISES expressed F.dr F₁ field F follows grad gradient h₁ h₂ Hence isotropic Laplace's equation line integral notation open region origin orthogonal orthonormality parallel parametric equation partial derivatives perpendicular plane Poisson's equation position vector proof prove rectangular cartesian coordinates relative respectively rotation scalar field scalar invariant scalar product second order tensor Show simple closed surface Solution sphere spherical polar coordinates suffixes transformation unit normal unit tangent unit vector vector field vector product verify x-axis x²+y²+x² xy-plane zero ди дх ду