In Lecture 6 we will look at combining these vector operators. For example, if your real number is represented by the letter t, the vector valued function might return [ h (t), g (t), f (t)] Vectors can be added to each other and multiplied by scalars. If ~ais a non-zero vector, the vector 1 j~aj ~ais the unique unit vector pointing in the same direction as ~a. In vector (or multivariable) calculus, we will deal with functions of two or three variables (usuallyx,yorx,y,z, respectively). Vector Calculus ... Collapse menu 1 Analytic Geometry. A unit vector is a vector of length one. Newton’s equation then takes the form m�= �1+�2, where �is the acceleration vector of the mass. 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. A simple example is a mass m acted on by two forces �1and �2. Here we find out how to. calculus. 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 Obvious examples are velocity, acceleration, electric field, and force. A vector function (also called a vector-valued function) takes in a real number as input and returns a vector. 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). The vector F is a gradient: F = ∇ 1 p x2+ y2+z2, (16.1.1) which turns out to be extremely useful. Distance Between Two Points; Circles 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). , 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”. 1. Examples of vectors are velocity and acceleration; examples of scalars are mass and charge. The underlying physical meaning — that is, why they are worth bothering about. Exercises 16.1. 16. 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 Find the unit vector ~upointing in the same direction as ~a= 4~i 2~j+~k. Example 3. 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. Lines; 2.