Hi there. Vector in the previous section We deal with functions and their derivatives. Or geometric sense of space curves We deal with derivatives. The following question may come to mind:Since you There are two main subject of the differential calculus. One integral derivative one. I wonder if the integral of vector functions Does it have a meaning? The answer is yes. There is the sense of vector functions. But these further issues will be examined. This multi-variable functions belong to the second them on the skin, the second course of their We will examine. Here we will only make an entry. Because it's something that's integral derivatives Does not always the question may come to mind. This will give the answer. We saw a space curve along a line or in a plane curve. To simplify the shapes in the plane Let's start. A point b to the point a Whether the curve. Above this, along the curve Whether defined size. Or, for example, electric charge mass As to the weight Or quantities such as temperature Suppose you have defined. What is the total mass we? Or, on which the total electricity What is the total heat load, or else What this curve to find answers to their questions we need to calculate an integral over. This is a numerical quantity to the integral leads us to. A gas flow a second embodiment, an water, a liquid flow Or electricity to the area in the E stream, and on it, let us consider a Let the curve. A space curve. The tangent of the curve on this and other units Let us consider the vector. Here the curve reserved. Here is a current. On this tangent vector of the current the projection and its accumulation to calculate the many problems encountered in bi issue. Or other vectors of this vector to the boundary their sum onto the take be required. For example, this line separated by a boundary that By the side of the mass exceed, business, mass flow vertical with component or this one, Or on this wire example of the current through the tube electric or sum of the currents find a liquid we may ask. This integration are needed. These can be grouped into three scopes as follows linear integrals. The first is the sum of numerical functions. Here is a gr e at each point, for example, intensity electric charge, a temperature which function to get the numeric values. The curve x t, y t, z is defined in t'yl Find, we find d, d to t h at d t'yl to divide again If the anchor. d, p divides d t were calculated. Wherein g is in the x, y and z t is the as a function of sheep here t is a function of a size of completely and interests that is in turn integral with respect to t we know we're coming to the integral single storey. A magnitude of the second vector that these space x, y, z will be defined at the point. t tangent on a unit vector on the If we take the projection again The function of the curve equation x t, y t is a function of the z function of t, t he t function is still divide to t d As you can see, if you multiply everything in t'yl Remove as a function of t. When this integral calculations, this pipe Or wire We have calculated the current flowing on. Bi in the direction perpendicular to the flow from one side to the other side n transitions to calculate the vector are projected onto. If you look here alone x, y, z saying here and you put only x. Because on a curve in space with n can not be described in b. Here we are now satisfied as in the plane. We will see later in this space two In order to separate the surface area needed. Normal vector to this surface on either side of needed. But surfaces also have not studied yet. For this reason, only two-dimensional form here promise. Things to do the same here x t, y is positioned as a function of t. he function of t. s also divide d d d with s to t, is here only t gets hit in t'yl function occurs. Remove this integral. As you can see with them, with this size numeric or vector Or along the tangent of the vertical size of e, In your drawing, sewing, e, in accordance with savings are calculated. Why you're getting is with s? Because these variables along a curve defined. For this reason, along the curve of the total 're getting. These curves open curve is possible, up to B when the sum of the sizes properly locate the closed base exits through curves. Again, this is likewise a digital the size of the or of a vector field tangent or perpendicular Your email address, the totals in the direction perpendicular to be required. These are very important issues. But here only to introduce this topic We are satisfied. From a mathematical perspective, that question also exposed I wanted to stay. We tried, but we deal with derivatives differential account both the derivative and integral requires. So what will be integral? Is there such a thing mean? If he can be defined to mind questions may occur. Yes, and many also are important issues. But we shall see in later sections. He introduced the subject by making an entry here I'm going. After that, no we want to do with the curves in space We finished threads. Thereafter, with functions of two variables I will try. The function of two variables What we mean by z, x and y change. These geometric event, as in space Shows surface. With two degrees of freedom because is a surface geometric objects. If the third move in the direction of the surface leave. So the two surfaces, only free variables to remain on the surface, though difficulty spots. How is that a single variable vector x terms When we define a space curve We have obtained puffin. These two variables related to surfaces before starting then it was generalized to three and n variables, We will generalize. But first, what these types of surfaces such basic surfaces are important? We'll start seeing them. Our next lesson on this surface issues to meet again to begin I hope. Goodbye.