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The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3. It can be defined in several ways, to be mentioned below:
An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
Q(x) divided by x(x + k) is a quadratic polynomial. Biquadratic equations. A quartic equation where a 3 and a 1 are equal to 0 takes the form + + = and thus ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
That is, Q kℓ is an identity matrix except for four entries, two on the diagonal (q kk and q ℓℓ, both equal to c) and two symmetrically placed off the diagonal (q kℓ and q ℓk, equal to s and −s, respectively), where c = cos θ and s = sin θ for some angle θ.
The Deduction Theorem for Q 0 shows that proofs from hypotheses using Rule R′ can be converted into proofs without hypotheses and using Rule R. Unlike some similar systems, inference in Q 0 replaces a subexpression at any depth within a WFF with an equal expression. So for example given axioms: 1. ∃x Px 2. Px ⊃ Qx