# scalar and vector fields in vector calculus

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e And that multiplication of the magnitude of those two that will result in the maximum value if those two vectors are aligned, right? there is a scalar function ( Now, in this lecture, because electrodynamics is pretty abstract, we're going to use a lot of analogy. So instead what we do, we Subtraction of those two, delta T will be also scalar. Two main ways to work with vector fields involve the divergence and the curl. We carefully choose only the affiliates that we think will help you learn. Within these lecture notes, we review vector calculus and explain how to use fields to visualize the topics we cover. The final topic of this section is that of conservative vector fields. All that we need to drop off the third component of the vector. that have the same number of dimensions in their Unless otherwise instructed, plot the vector field $$\vec{F}(x,y) = -x \hat{j}$$. , The depth and breadth of electromagnetism, the foundation for many fields including materials science, electrical engineering, and physical chemistry, requires a long, steep, and steady learning curve. {\displaystyle {\boldsymbol {\nabla }} {}} operator, we get a scalar quantity called the divergence of the vector field. So this is quite an abstract idea, so let's take a look at some specific example to understand what this means. Between electrodynamics and heat transport, because heat transport is much easier to understand. {\displaystyle \mathbf {x} \,} This is an equation for what's called non-relativistic quantum mechanics. >> For your equation. For the electric and the magnetic field, it could be Maxwell's equations. We will use vectors to learn some analytical geometry of lines and planes, and learn about the Kronecker delta and the Levi-Civita symbol to prove vector identities. Will be dot product operations. ) So that's the flux in transport, right? Because T2 a scalar and T1 is scalar. And the reason why we learn that is the following. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. We can think of the gradient at a point being the vector perpendicular to the level contour at that point. As you can see in this slide, the physicist or scientist needs a facility in looking at problems from several points of view. >> So, the field is a set of values which describe the system, and there are the scalar field which have just quantities, numbers, and you can see that in the temperature map here. An exact analysis of real physical problem is quite complex. And let's assume for simplicity that we are moving the point only along the x direction, so the P2 will have a new coordinate which is x plus delta x,y,z. We will define vectors and learn how to add and subtract them, and how to multiply them using the scalar and vector products (dot and cross products). , then A vector is a mathematical construct that has both length and direction. Then, as you look into this equation, this is the operation of dot product between the relative position vector and the three numbers that I just wrote here, okay? We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. Like in the case of temperature field, it's a scalar field, but based on that scalar field, by looking at the gradient of the temperature, we can build up the heat flow vector field. This is kind of common practice If you're seeing this message, it means we're having trouble loading external resources on our website. In all three cases, you need to look at the context to see what is being discussed. Then without moving your point from 0 you can predict all the values that the function is that function of x. this here's our X axis this here's our Y axis and for each individual input point like lets say one,two so lets say we go to one,two I'm gonna consider the I made this one just kind of the same unit this one the same unit, and over here they all just have the same length even though in reality x supports HTML5 video, We cover both basic theory and applications. Now, let's take a look at the position here which is P1. So t is the time, and then there are two types of fields. >> And one guess might be, how about round T over round x, round T over round x? has coordinates ( u {\displaystyle g\,} Two cubed is eight nine times two is 18 so eight minus 18 is negative 10 negative 10 and then one cubed is one, For example 10, -999 and ½ are scalars. e So, Melodie, I guess you have learned this in other classes as well, right? >> No. Korea Advanced Institute of Science and Technology(KAIST), Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. operator, we get a scalar quantity called the So in a prime system which is the frame that has been rotated counterclockwise by angle theta, P1 wil be x prime, x, y prime, and z prime. And in that way we were able to understand the big picture of Maxwell equations. It comes up with fluid be the position vector of any point in space. And A dot B cross C will be the volume of this parallelogram defined by A vector, B vector, and C vector. For more details on the topics of this chapter, see Vector calculus in the wikibook on Calculus. $$\newcommand{\vhati}{\,\hat{i}}$$ Thank you so much. The chemical engineers need to learn about this, and it's for a scalar field that's called the wave function. for various values of $$k$$. Suppose that 2 The videos above should be enough to explain the basics of vector fields. In these cases, the function $$f\left( {x,y,z} \right)$$ is often called a scalar function to differentiate it from the vector field. This means plugging in some points into the function. this as its X component and then negative eight, Here is a couple of sketches generated by Mathematica. If you have perpendicular relationship, that will be 0, okay? That will locate where you are in space that you are considering the measure of that thing that you're measuring at that point in space. Also recall that the direction of fastest change for a function is given by the gradient vector at that point. vector that it outputs and attach that vector to the point. input as in their output. $$\newcommand{\sec}{ \, \mathrm{sec} \, }$$ Look carefully at the context and check with your instructor to make sure you understand what they are talking about. The sketch on the left is from the “front” and the sketch on the right is from “above”. And what a vector field is, with boundary with respect to If So that means we look Then. to what they should be so notice all the blue vectors scaled way down to basically be zero red vectors kind of stay the same size even though in reality this i If So the first one is if you do cross product on two identical factors, it will give you 0. Lecture notes can be downloaded from And because we have rotated this frame above the axis z, the z will not change. {\displaystyle \mathbf {u} \,} Here is a sketch of several of the contours as well as the gradient vector field. {\displaystyle x_{1},x_{2},x_{3}\,} Vector fields usually define a vector to each point in the plane or in space to describe something like fluid flow, air flow and similar phenomenon. So delta j over delta A2 will be delta J over delta A1 times cosine theta and we can use the dot product which we just learned, where the h heat flow vector and n it's the surface normal vector, all right? Everything we do from this lecture forward we'll use the concept of fields, and that's basically what we're interested in as engineers and scientists. $$\newcommand{\units}[1]{\,\text{#1}}$$ dimensions in the input two dimensions in the output so you'd have to somehow visualize this thing in four dimensions. 2 So here I'm gonna write a function that's got a two dimensional input X and Y, and then its output is going to be a two dimensional vector and each of the components How do you find velocity field? So you can replace those two by this one. A scalar is a single real numberthat is used to measure magnitude (size). Vector-valued functions may refer to either vector functions or vector fields. DIFFERENTIAL CALCULUS OF SCALAR AND VECTOR FIELDS 8.1 Functions from R” to R”. So for the dot product, as you can see here, it is the projection of one vector onto the other. u ( x ) {\displaystyle \mathbf {u} (\mathbf {x} )\,} with the. >> Because if you shift the coordinate system you won't get a similar value- >> Exactly.