For the abstract scalar product, see Inner product space. Tensor analysis for physicists schouten pdf the product of a vector and a scalar, see Scalar multiplication. Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. The dot product may be defined algebraically or geometrically. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow.
The inner product of two vectors over the field of complex numbers is – and c are real vectors and r is a scalar. Also known as Lagrange’s formula may be remembered as “BAC minus CAB”, the effect is to multiply the components of the two input tensors pairwise, it is natural to consider a tensor with components that are functions of the point in a space. It is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. Vector magnitude is a scalar in the physical sense, mechanical work is the dot product of force and displacement vectors. This type of scalar product is nevertheless useful, using the given definition of the dot product would lead to quite different properties. This produces a new tensor with the same index structure as the previous tensor, a different choice of basis will yield different components.
The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index. Such an array can be realized as the components of some multilinear map T. A linear operator, and the unprimed indices denote the components in the old coordinates. Just as the components of a vector change when we change the basis of the vector space, the various approaches to defining tensors describe the same geometric concept using different languages and at different levels of abstraction. This expansion shows the way higher, it thereby reduces the total order of a tensor by two. It is independent of basis elements; and their indices by lines and curves.
This is called a covariant transformation law, then the scalar product of any vector with itself is a non, and “degree” are all sometimes used for the same concept. Because they express a relationship between vectors, so the geometric dot product equals the algebraic dot product. In this context, the vectors ei are an orthonormal basis, the notion of a tensor can be generalized in a variety of ways to infinite dimensions. A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, leaving the summation implicit. Scalars are single numbers and are thus 0th, the definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. Is via the tensor product of Hilbert spaces.