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**Extra info for Schaum's Outline of Mathematical Handbook of Formulas and Tables, 4th Edition: 2,400 Formulas + Tables (Schaum's Outlines)**

E. , a right-threaded screw circled via an attitude below one hundred eighty° from A to B will improve towards u as in Fig. 20-5). primary effects stick with: i A × = A B 20. 12. 1 B1 j A2 B2 okay A3 B3 u B A = ( A2 B3 − A3 B2 )i + ( A3 B1 − A1 B3 ) j + ( A1 B2 − A2 B1 )k Fig. 20-5 20. thirteen. A × B = − (B × A) 20. 14. A × (B + C) = A × B + A × C 20. 15. | A × B | = region of parallelogram having aspects A and B Miscellaneous formulation regarding Dot and pass items 20. sixteen. A1 A2 A i (B × C) = B1 B2 C1 C2 A3 B3 = A1 B2C3 + A2 B3C1 + A3 B1C2 − A3 B2C1 − A2 B1C3 − A1 B3C2 C3 20. 17. | A • (B × C) | = quantity of parallelepiped with aspects A, B, C 20. 18. A × (B × C) = B(A • C) − C(A • B) 20. 19. (A × B) × C = B(A • C) − A(B • C) 20. 20. (A × B) • (C × D) = (A • C)(B • D) − (A • D)(B • C) 20. 21. (A × B) × (C × D) = C{A • (B × D)}− D{A • (B × C)} = B{A • (C × D)} − A{B • (C × D)} FORMULAS FROM VECTOR research 122 Derivatives of Vectors The by-product of a vector functionality A(u) = A1(u)i + A2(u)j + A3(u)k of the scalar variable u is given by way of 20. 22. dA dA dA A(u + Δu) − A(u) dA1 = lim = i+ 2 j+ three okay Δ u → zero du Δu du du du Partial derivatives of a vector functionality A(x, y, z) are equally outlined. We think that each one derivatives exist until differently distinct. formulation related to Derivatives 20. 23. d dB dA (A i B) = A i + iB du du du 20. 24. d dB dA (A × B) = A × + ×B du du du 20. 25. ⎞ ⎛ dB ⎛ dA d dC ⎞ {A i (B × C )} = i (B × C ) + A i ⎜ × C⎟ + A i ⎜ B × du du du ⎟⎠ ⎠ ⎝ du ⎝ 20. 26. Ai dA dA =A du du 20. 27. Ai dA =0 du if | A | is a continuing The Del Operator The operator del is outlined by way of 20. 28. ∇=i ∂ ∂ ∂ + j +k ∂z ∂y ∂x within the following effects we think that U = U(x, y, z), V = V(x, y, z), A = A(x, y, z) and B = B(x, y, z) have partial derivatives. The Gradient ⎛ ∂ ∂U ∂U ∂U ∂⎞ ∂ i+ j+ okay 20. 29. Gradient of U = grad U = ∇U = ⎜ i + j + okay ⎟ U = ∂x ∂y ∂z ∂z ⎠ ∂y ⎝ ∂x The Divergence ⎛ ∂ ∂ ∂⎞ 20. 30. Divergence of A = div A = ∇ • A = ⎜ i + j + ok ⎟ i ( A1 i + A2 j + A3 okay) ∂y ∂z ⎠ ⎝ ∂x = ∂A1 ∂A2 ∂A3 + + ∂x ∂y ∂z FORMULAS FROM VECTOR research The Curl 20. 31. Curl of A = curl A = ∇ × A ⎛ ∂ ∂ ∂⎞ = ⎜ i + j + okay ⎟ × ( A1 i + A2 j + A3 ok) ∂ ∂ ∂ x y z⎠ ⎝ i ∂ = ∂x A1 j ∂ ∂y A2 ok ∂ ∂z A3 ⎛ ∂A ∂A ⎞ ⎛ ∂A ∂A ⎞ ⎛ ∂A ∂A ⎞ = ⎜ three − 2 ⎟ i + ⎜ 1 − three⎟ j + ⎜ 2 − 1⎟ ok ∂z ⎠ ∂y ⎠ ∂x ⎠ ⎝ ∂x ⎝ ∂y ⎝ ∂z The Laplacian 20. 32. Laplacian of U = ∇ 2U = ∇ i (∇U ) = 20. 33. Laplacian of A = ∇ 2 A = ∂ 2U ∂ 2U ∂ 2U + + 2 ∂x 2 ∂y 2 ∂z ∂2 A ∂2 A ∂2 A + 2 + 2 ∂z ∂y ∂x 2 The Biharmonic Operator 20. 34. Biharmonic operator on U = ∇ 4U = ∇ 2 (∇ 2U ) = ∂ 4U ∂ 4U ∂ 4U ∂ 4U ∂ 4U ∂ 4U + + + + 2 + 2 2 ∂x 2 ∂y 2 ∂z four ∂x 2 ∂z 2 ∂y four ∂y 2 ∂z 2 ∂x four Miscellaneous formulation concerning ∇ 20. 35. ∇(U + V ) = ∇U + ∇V 20. 36. ∇ i (A + B) = ∇ i A + ∇ i B 20. 37. ∇ × (A + B) = ∇ × A + ∇ × B 20. 38. ∇ i (UA) = (∇U ) i A + U (∇ i A) 20. 39. ∇ × (UA) = (∇U ) × A + U (∇ × A) 20. forty. ∇ i (A × B) = B i (∇ × A) − A i (∇ × B) 20. forty-one. ∇ × (A × B) = (B i ∇) A − B(∇ i A) − (A i ∇)B + A(∇ i B) 20. forty two. ∇(A i B) = (B i ∇)A + (A i ∇)B + B × (∇ × A) + A × (∇ × B) 20.