It is a representation of tensors.
The sum of its diagonal elements, represented as tr A = A11 + A22 + A33.
Kinematics.
2 = (G), (2) 20 2 = 18 /10 10 1 - 10 I 3.
Numerical methods applicable to fluid mechanics.
Using permutation and indices for simplification.
A mathematical object that can be represented as a matrix and relates vectors in a linear transformation.
Jensors.
It indicates the gradient of a scalar function.
E_ijk = 1.
Nabla operator (∇).
It represents a relationship involving the Kronecker Delta and indices.
123, 231, 312.
A mathematical object that can be represented as a vector in a given coordinate system.
A measure of the length of a vector in Euclidean space.
It implies summation of that term over all the values of the index.
It refers to the gradient operator applied to a function.
A mathematical object that can be represented as a linear mapping in continuum mechanics.
Tensor Calculus.
A second-order tensor.
It serves as a tool for simplifying tensor equations and represents the identity tensor.
It is used to describe relationships between vectors and tensors.
The determinant must be non-zero (det ≠ 0).
Basic Conservation Equations.
a · b = a_i b_j S_ij, where S_ij is the metric tensor.
It provides a systematic way to represent and manipulate tensors.
A third-order tensor used in tensor calculus.
It represents a specific condition or equation involving the tensor Z.
ab = E_ijk a_i b_j.
A function that is 1 if the indices are equal and 0 otherwise.
It indicates a specific relationship or value associated with the tensor A.
It provides important information about the stress or strain state of a material.
tr A = Aij eij, where eij represents the basis vectors.
321, 132, 213.
The gradient increases the order of a tensor.
As a linear mapping represented by the equation 3 = DijDij.
It typically represents components of the tensor in a specific direction.
It refers to the simplest form of a tensor, which can be increased in order through operations like the gradient.
It is used to determine the invertibility of the tensor.
It is represented as Aji = Aij, indicating the relationship between the elements of the tensor.
Scalar Problems and Fluid Mechanical Problems.
It indicates a specific component of a tensor.
Cartesian coordinates (Xi).
It denotes the double dot product between two tensors A and B.
det E_ijk = -1.
The derivative of the components of a tensor.
The Cartesian representation of a tensor.
It is denoted as a = 1.5, b = A, indicating the relationship between the original tensor and its inverse.
A vector expressed in terms of its components and basis vectors.
