Vector Calculus, Fourth Edition, makes use of the language and notation of vectors and matrices to educate multivariable calculus. it really is excellent for college students with a great history in single-variable calculus who're in a position to considering in additional normal phrases in regards to the subject matters within the path. this article is exceptional from others by means of its readable narrative, a number of figures, thoughtfully chosen examples, and punctiliously crafted workout units. Colley contains not just uncomplicated and complex routines, but in addition mid-level routines that shape an important bridge among the 2.
Vector Calculus by Susan Jane Colley
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Y intersect the plane 2x − 3y + z = 7? 41. Let a, b, c be nonzero constants. Show that the line with parametric equations x = at + a, y = b, z = ct + c lies on the surface with equation x 2 /a 2 + y 2 /b2 − z 2 /c2 = 1. 2π 4π x 42. Find the point of intersection of the two lines l1 : x = 2t + 3, y = 3t + 3, z = 2t + 1 and l2 : x = 15 − 7t, y = t − 2, z = 3t − 7. 43. Do the lines l1 : x = 2t + 1, y = −3t, z = t − 1 and l2 : x = 3t + 1, y = t + 5, z = 7 − t intersect? Explain your answer. 32 A curtate cycloid.
A · a ≥ 0, and a · a = 0 if and only if a = 0; a · b = b · a; a · (b + c) = a · b + a · c; (ka) · b = k(a · b) = a · (kb). Proof of Property 1 If a = (a1 , a2 , a3 ), then we have a · a = a1 a1 + a2 a2 + a3 a3 = a12 + a22 + a32 . This last expression is evidently nonnegative, since it is a sum of squares of real numbers. Moreover, such an expression is zero exactly when each of the terms is ■ zero, that is, if and only if a1 = a2 = a3 = 0. We leave the proofs of properties 2, 3, and 4 as exercises.
In physics, when a constant force acts on an object as the object is displaced, the work done by the force is the product of the length of the displacement and the component of the force in the direction of the displacement. 48 depicts an object acted upon by a constant force F, which displaces it from the point P to the point Q. Let θ denote the angle between F and the direction of displacement. (a) Show that the work done by F is determined by the −→ formula F · P Q. (b) Find the work done by the (constant) force F = i + 5j + 2k in moving a particle from the point (1, −1, 1) to the point (2, 0, −1).
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