Prove using Levi-Civita veetors: (A)x B) . (C × D) = (A . C)(B . D)...
Prove using Levi-Civita veetors: (A × B) . (C x D) = (A-C)(B . D) _ (A-D)(B-C)
Show that A*(BxC) = B*(CxA) = C*(AxB) Using the Levi Civita symbol (A,B,C are all vectors).
4. You saw in class how the curl of a vector can be expressed using the Levi-Civita symbol. Another way to write the curl of a vector is as the deter- minant of a matrix with the basis vectors in the top row, partial derivatives in the second and vector components in the third. It should therefore not surprise you that the determinant of any 3 x 3 matrix can be written using the Levi-Civita symbol. a. Write the determinant...
Using Cartesian tensor notation and the fact that we can write the curl operator in the form: (vxA), = eijk BKI where Eijk is the Levi-Civita tensor, and the relationship: Ekijeklm = dilim - Sim Oil prove Greens' vector identity: 02(A. B) = A -12B – B-02Ã+20. [(B.V)X + B ® (V x A)] where A and B are vector functions. [8 marks]
Prove that B = {(a,b) x (c,d) | a,b,c,d EQ, a<b, c<d} is a basis for some topology on R2.
t5.14. Let x and y be two different coordinate patches for part of a surface M Let X Xx, X, and Y = Y'x Y'y. be two vector fields. Define symbols Z* and Z by χι ΣΓ/Υιχ | and Prove that 2 EZ*(dv|8u*). (Hint: Problem 4.11.) This proves thatZx £ Zy, defines a vector field Z = VxY, called the covariant derivative of Y with respect to X. This is one of the most fundamental concepts of modern differential geometry....
first picture is the notation on the test paper Second picture is the question i would like to solve thanks~ Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
Prove that: A'+B'+C'+D' = A'B'C'D' using theorems of boolean algebra to prove DeMorgans theorem for four variables
4. Prove that if A-B-C and A-C-B, then A-B-C-D 4. Prove that if A-B-C and A-C-B, then A-B-C-D
2. Prove that A+B AB by: a. b. c. d. Using truth tables for both the right and right sides of the equation. Drawing a gate level schematic for both the right and right sides of the equation Which theorem is this? Restate the theorem in terms of gates.