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Overview. Many of you will know a good deal already about Vector Algebra — how to add and subtract vectors, how to take scalar and vector products of vectors, and something of how to describe geometric and physical entities using vectors.
The unit vectors along the x,y,z axes in Cartesian coordinates are ^i,^j,^k. Vectors can be added geometrically, by placing them end-to-end to see the resultant vector (diagram above right), or they can be added algebraically (add the vector components): F + G = (FX + GX) i + (FY + GY) j + (FZ + GZ) k
These notes are meant to be a support for the vector calculus module (MA2VC/MA3VC) taking place at the University of Reading in the Autumn term 2016. The present document does not substitute the notes taken in class, where more examples and proofs are provided and where the content is discussed in greater detail.
15.1 Vector Fields. For an ordinary scalar function, the input is a number x and the output is a number. f .x/: For a vector field (or vector function), the input is a point .x;y/ and the output is a two-dimensional vector F.x;y/: There is a “field” of vectors, one at every point.
A three-dimensional vector field has components M(x, y, z) and N(x, y, z) and P(x, y, 2). Then the vectors are F = Mi + Nj + Pk. EXAMPLE 1 The position vector at (x, y) is R = xi + yj. Its components are M = x and N = y. The vectors grow larger as we leave the origin (Figure 15.la).
Vector Calculus. In this chapter we develop the fundamental theorem of the Calculus in two and three dimensions. This begins with a slight reinterpretation of that theorem. Consider the endpoints a b of the interval a b from a to b as the boundary of that interval. Then the fundamental theorem, in this form:
its unit vector is referred to by e v or v^ which is equal to ^v = v=kvk. One would say that the unit vector carries the information about direction. Therefore magnitude and direction as constituents of a vector are multiplicatively decomposed as v= vv^. 1.4 Vector decomposition Writing a vector vas the sum of its components is called ...
