In vector (or multivariable) calculus, we will deal with functions of two or three variables (usuallyx,yorx,y,z, respectively). The underlying physical meaning — that is, why they are worth bothering about. If ~ais a non-zero vector, the vector 1 j~aj ~ais the unique unit vector pointing in the same direction as ~a. For example, if your real number is represented by the letter t, the vector valued function might return [ h (t), g (t), f (t)] 5.1 The gradient of a scalar ﬁeld Recall the discussion of temperature distribution throughout a room in the overview, where we wondered how a scalar would vary as we moved off in an arbitrary direction. The important vector calculus formulas are as follows: From the fundamental theorems, you can take, F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k Fundamental Theorem of the Line Integral Vectors can be diﬀerentiated component by component: F(u) = Fi(u)ei⇒F′(u) = F′ i(u)ei provided the basis {ei}is independent of u(note the implicit summation convention over repeated indices). In Lecture 6 we will look at combining these vector operators. The vector F is a gradient: F = ∇ 1 p x2+ y2+z2, (16.1.1) which turns out to be extremely useful. A unit vector is a vector of length one. Distance Between Two Points; Circles j~aj= p 4 2+ ( 2)2 + 1 = p 21; so ~u= 1 p 21 ~a= 4 p 21 ~i+ 2 p 21 ~j+ 1 p 21 ~k 1.3 Dot Product 16. Obvious examples are velocity, acceleration, electric ﬁeld, and force. Vectors can be added to each other and multiplied by scalars. Example 3. Below, prob- ably all our examples will be of these “magnitude and direction” vectors, but we should not forget that many of the results extend to the wider realm of vectors. Here we ﬁnd out how to. The graph of a function of two variables, say,z=f(x,y), lies inEuclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b,c). A vector function (also called a vector-valued function) takes in a real number as input and returns a vector. Find the unit vector ~upointing in the same direction as ~a= 4~i 2~j+~k. Examples of vectors are velocity and acceleration; examples of scalars are mass and charge. , which is the reciprocal of the square of the distance from (x,y,z) to the origin—in other words, F is an “inverse square law”. Exercises 16.1. Vector Calculus ... Collapse menu 1 Analytic Geometry. Newton’s equation then takes the form m�= �1+�2, where �is the acceleration vector of the mass. 1. calculus. Lines; 2. A simple example is a mass m acted on by two forces �1and �2.

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