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Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
Their resultant is defined as the element of k obtained by replacing in the generic resultant the indeterminate coefficients by the actual coefficients of the . The main property of the resultant is that it is zero if and only if P 1 , … , P n {\displaystyle P_{1},\ldots ,P_{n}} have a nonzero common zero in an algebraically closed extension ...
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field T {\displaystyle \mathbf {T} } of non-zero order k is written as div ( T ) = ∇ ⋅ T {\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} } , a contraction of a tensor field ...
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces. [1]
The resultant vector is invariant of rotation of basis. Due to the dependence on handedness, the cross product is said to be a pseudovector. In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space.
Calculus serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notably physics and engineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. This is because the n-dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors.