By Kwong-Tin Tang

Pedagogical insights won via 30 years of training utilized arithmetic led the writer to write down this set of student-oriented books. issues comparable to advanced research, matrix concept, vector and tensor research, Fourier research, fundamental transforms, traditional and partial differential equations are awarded in a discursive sort that's readable and straightforward to persist with. quite a few truly said, thoroughly labored out examples including conscientiously chosen challenge units with solutions are used to augment scholars' figuring out and manipulative ability. The aim is to make scholars cozy and assured in utilizing complicated mathematical instruments in junior, senior, and starting graduate courses.

**Read or Download Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms (v. 2) PDF**

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**Additional info for Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms (v. 2)**

**Example text**

When it operates on a scalar function, it turns ∇ϕ into a vector with deﬁnite magnitude and direction. It also has a deﬁnite physical meaning. 1. Show that ∇r = r and ∇f (r) = rdf /dr, where r is a unit vector along the position vector r = xi + yj + zk and r is the magnitude of r. 1. ∂ ∂ ∂ +j +k r, ∂x ∂y ∂z ∂ ix ix ∂r 1/2 =i x2 + y 2 + z 2 = = , i 1/2 2 2 2 ∂x ∂x r (x + y + z ) xi + yj + zk r ix jy kz + + = = = r. ∇r = r r r r r ∇r = i etc. ∂f ∂f ∂f +j +k , ∂x ∂y ∂z ∂f df ∂r df x i =i =i , etc.

10i + 3j + 11k, −10i − 3j − 11k. 8. Find the area of a parallelogram having diagonals A = 3i + j − 2k and B=i− √ 3j + 4k. Ans. 5 3. 9. Evaluate (2i − 3j) · [(i + j − k) × (3j − k)]. Ans. 4. 10. Find the volume of the parallelepied whose edges are represented by A = 2i − 3j + 4k, B = i + 2j − k, and C = 3i − j + 2k. Ans. 7. 11. Find the constant a such that the vectors 2i − j + k, i + 2j − 3k and 3i + aj + 5k are coplanar. Ans. a = −4. 12. Show that (a) (b × c) × (c × a) = c (a · b × c) ; 2 (b) (a × b) · (b × c) × (c × a) = (a · b × c) .

To ﬁnd the perpendicular distance between them is a diﬃcult problem in analytical geometry. With vectors, it is relatively easy. Let the equations of two such lines be r = r1 + tv1 , r = r2 + t v2 . 3 Lines and Planes 25 Let a on line 1 and b on line 2 be the end points of the common perpendicular on these two lines. 47) with t = t2 . Accordingly ra = r1 + t1 v1 , rb = r2 + t2 v2 . 49) Since rb − ra is perpendicular to both v1 and v2 , it must be in the direction of (v1 × v2 ) . If d is the length of rb − ra , then rb − ra = v1 × v2 d.