Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.
Vector and tensor algebra in three-dimensional curvilinear coordinates
Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna. Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi, Simmonds, Green and Zerna, Basar and Weichert, and Ciarlet.
Coordinate transformations Consider two coordinate systems with coordinate variables ( Z 1 , Z 2 , Z 3 ) {\displaystyle (Z^{1},Z^{2},Z^{3})} and ( Z 1 ´ , Z 2 ´ , Z 3 ´ ) {\displaystyle (Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})} , which we shall represent in short as just Z i {\displaystyle Z^{i}} and Z i ´ {\displaystyle Z^{\acute {i}}} respectively and always assume our index i {\displaystyle i} runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates Z i {\displaystyle Z^{i}} and Z i ´ {\displaystyle Z^{\acute {i}}} may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates Z i {\displaystyle Z^{i}} and Z i ´ {\displaystyle Z^{\acute {i}}} are functions of each other
Z i = f i ( Z 1 ´ , Z 2 ´ , Z 3 ´ ) {\displaystyle Z^{i}=f^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})} for i = 1 , 2 , 3 {\displaystyle i=1,2,3}
which can be written as
Z i = Z i ( Z 1 ´ , Z 2 ´ , Z 3 ´ ) = Z i ( Z i ´ ) {\displaystyle Z^{i}=Z^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})=Z^{i}(Z^{\acute {i}})} for i ´ , i = 1 , 2 , 3 {\displaystyle {\acute {i}},i=1,2,3}
These three equations together are also called a coordinate transformation from Z i ´ {\displaystyle Z^{\acute {i}}} to Z i {\displaystyle Z^{i}} . Let us denote this transformation by T {\displaystyle T} . We will therefore represent the transformation from the coordinate system with coordinate variables Z i ´ {\displaystyle Z^{\acute {i}}} to the coordinate system with coordinates Z i {\displaystyle Z^{i}} as:
Z = T ( z ´ ) {\displaystyle Z=T({\acute {z}})}
Similarly we can represent Z i ´ {\displaystyle Z^{\acute {i}}} as a function of Z i {\displaystyle Z^{i}} as follows:
Z i ´ = g i ´ ( Z 1 , Z 2 , Z 3 ) {\displaystyle Z^{\acute {i}}=g^{\acute {i}}(Z^{1},Z^{2},Z^{3})} for i ´ = 1 , 2 , 3 {\displaystyle {\acute {i}}=1,2,3}
and we can write the free equations more compactly as
Z i ´ = Z i ´ ( Z 1 , Z 2 , Z 3 ) = Z i ´ ( Z i ) {\displaystyle Z^{\acute {i}}=Z^{\acute {i}}(Z^{1},Z^{2},Z^{3})=Z^{\acute {i}}(Z^{i})} for i ´ , i = 1 , 2 , 3 {\displaystyle {\acute {i}},i=1,2,3}
These three equations together are also called a coordinate transformation from Z i {\displaystyle Z^{i}} to Z i ´ {\displaystyle Z^{\acute {i}}} . Let us denote this transformation by S {\displaystyle S} . We will represent the transformation from the coordinate system with coordinate variables Z i {\displaystyle Z^{i}} to the coordinate system with coordinates Z i ´ {\displaystyle Z^{\acute {i}}} as:
z ´ = S ( z ) {\displaystyle {\acute {z}}=S(z)}
If the transformation T {\displaystyle T} is bijective then we call the image of the transformation, namely Z i {\displaystyle Z^{i}} , a set of admissible coordinates for Z i ´ {\displaystyle Z^{\acute {i}}} . If T {\displaystyle T} is linear the coordinate system Z i {\displaystyle Z^{i}} will be called an affine coordinate system, otherwise Z i {\displaystyle Z^{i}} is called a curvilinear coordinate system.
The Jacobian As we now see that the Coordinates Z i {\displaystyle Z^{i}} and Z i ´ {\displaystyle Z^{\acute {i}}} are functions of each other, we can take the derivative of the coordinate variable Z i {\displaystyle Z^{i}} with respect to the coordinate variable Z i ´ {\displaystyle Z^{\acute {i}}} . Consider
∂ Z i ∂ Z i ´ = d e f J i ´ i {\displaystyle {\frac {\partial {Z^{i}}}{\partial {Z^{\acute {i}}}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;J_{\acute {i}}^{i}} for i ´ , i = 1 , 2 , 3 {\displaystyle {\acute {i}},i=1,2,3} , these derivatives can be arranged in a matrix, say J {\displaystyle J} , in which J i ´ i {\displaystyle J_{\acute {i}}^{i}} is the element in the i {\displaystyle i} -th row and i ´ {\displaystyle {\acute {i}}} -th column
J = ( J 1 ´ 1 J 2 ´ 1 J 3 ´ 1 J 1 ´ 2 J 2 ´ 2 J 3 ´ 2 J 1 ´ 3 J 2 ´ 3 J 3 ´ 3 ) = ( ∂ Z 1 ∂ Z 1 ´ ∂ Z 1 ∂ Z 2 ´ ∂ Z 1 ∂ Z 3 ´ ∂ Z 2 ∂ Z 1 ´ ∂ Z 2 ∂ Z 2 ´ ∂ Z 2 ∂ Z 3 ´ ∂ Z 3 ∂ Z 1 ´ ∂ Z 3 ∂ Z 2 ´ ∂ Z 3 ∂ Z 3 ´ ) {\displaystyle J={\begin{pmatrix}J_{\acute {1}}^{1}&J_{\acute {2}}^{1}&J_{\acute {3}}^{1}\\J_{\acute {1}}^{2}&J_{\acute {2}}^{2}&J_{\acute {3}}^{2}\\J_{\acute {1}}^{3}&J_{\acute {2}}^{3}&J_{\acute {3}}^{3}\end{pmatrix}}={\begin{pmatrix}{\partial {Z^{1}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{2}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{3}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {3}}}}\end{pmatrix}}}
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates Let ( b 1 , b 2 , b 3 ) {\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3})} be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v {\displaystyle \mathbf {v} } can be expressed as
v = v k b k {\displaystyle \mathbf {v} =v^{k}\,\mathbf {b} _{k}}
The components v k {\displaystyle v^{k}} are the contravariant components of the vector v {\displaystyle \mathbf {v} } . The reciprocal basis ( b 1 , b 2 , b 3 ) {\displaystyle (\mathbf {b} ^{1},\mathbf {b} ^{2},\mathbf {b} ^{3})} is defined by the relation
b i ⋅ b j = δ j i {\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}}
where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta. The vector v {\displaystyle \mathbf {v} } can also be expressed in terms of the reciprocal basis:
v = v k b k {\displaystyle \mathbf {v} =v_{k}~\mathbf {b} ^{k}}
The components v k {\displaystyle v_{k}} are the covariant components of the vector v {\displaystyle \mathbf {v} } .
Second-order tensors in curvilinear coordinates A second-order tensor can be expressed as
S = S i j b i ⊗ b j = S j i b i ⊗ b j = S i j b i ⊗ b j = S i j b i ⊗ b j {\displaystyle {\boldsymbol {S}}=S^{ij}~\mathbf {b} _{i}\otimes \mathbf {b} _{j}=S_{~j}^{i}~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}=S_{i}^{~j}~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}=S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}}
The components S i j {\displaystyle S^{ij}} are called the contravariant components, S j i {\displaystyle S_{~j}^{i}} the mixed right-covariant components, S i j {\displaystyle S_{i}^{~j}} the mixed left-covariant components, and S i j {\displaystyle S_{ij}} the covariant components of the second-order tensor.
Metric tensor and relations between components The quantities g i j {\displaystyle g_{ij}} , g i j {\displaystyle g^{ij}} are defined as
g i j = b i ⋅ b j = g j i ; g i j = b i ⋅ b j = g j i {\displaystyle g_{ij}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}=g_{ji}~;~~g^{ij}=\mathbf {b} ^{i}\cdot \mathbf {b} ^{j}=g^{ji}}
From the above equations we have
v i = g i k v k ; v i = g i k v k ; b i = g i j b j ; b i = g i j b j {\displaystyle v^{i}=g^{ik}~v_{k}~;~~v_{i}=g_{ik}~v^{k}~;~~\mathbf {b} ^{i}=g^{ij}~\mathbf {b} _{j}~;~~\mathbf {b} _{i}=g_{ij}~\mathbf {b} ^{j}}
The components of a vector are related by
v ⋅ b i = v k b k ⋅ b i = v k δ k i = v i {\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} ^{i}=v^{k}~\delta _{k}^{i}=v^{i}}
v ⋅ b i = v k b k ⋅ b i = v k δ i k = v i {\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} _{i}=v_{k}~\delta _{i}^{k}=v_{i}}
Also,
v ⋅ b i = v k b k ⋅ b i = g k i v k {\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} _{i}=g_{ki}~v^{k}}
v ⋅ b i = v k b k ⋅ b i = g k i v k {\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} ^{i}=g^{ki}~v_{k}}
The components of the second-order tensor are related by
S i j = g i k S k j = g j k S k i = g i k g j l S k l {\displaystyle S^{ij}=g^{ik}~S_{k}^{~j}=g^{jk}~S_{~k}^{i}=g^{ik}~g^{jl}~S_{kl}}
The alternating tensor In an orthonormal right-handed basis, the third-order alternating tensor is defined as
E = ε i j k e i ⊗ e j ⊗ e k {\displaystyle {\boldsymbol {\mathcal {E}}}=\varepsilon _{ijk}~\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}\otimes \mathbf {e} ^{k}}
In a general curvilinear basis the same tensor may be expressed as
E = E i j k b i ⊗ b j