An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar.A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T (v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the ...
Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped me to see how this rhythm plays its way throughout the various formalisms.
more. Second, tensor theory, at the most elementary level, requires only linear algebra and some calculus as prerequisites. Proceeding a small step further, tensor theory requires background in multivariate calculus. For a deeper understanding, knowledge of manifolds and some point-set topology is required.
Tensor. An th-rank tensor in -dimensional space is a mathematical object that has indices and components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of space.
Now, a tensor is the most general concept. Scalars, vectors, and matrices are all tensors of ranks 0, 1, and 2, respectively. Tensors are simply a generalization of the concepts we have seen so far. An object we haven’t seen is a tensor of rank 3. Its dimensions could be signified by k,m, and n, making it a KxMxN object.
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The simplest and most common case is when you attempt to multiply or add a tensor to a scalar. In that case, the scalar is broadcast to be the same shape as the other argument. x = tf.constant( [1, 2, 3]) y = tf.constant(2) z = tf.constant( [2, 2, 2]) # All of these are the same computation.
A Tensor is a mathematical object similar to, but more general than, a vector and often represented by an array of components that describe functions relevant to coordinates of a space. Put simply, a Tensor is an array of numbers that transform according to certain rules under a change of coordinates.