 Poitier Wright - Plane of a Normal Vector Poitier Wright - Doing my best to understand Mathematics Plane of a Normal Vector Functions concept. I have also done my best to write this in my own words with the following references. Thank you for watching. Locating the distance from a plane to it’s point is frequently necessary as is useful in the confirmation between lines and points as stated in my previous video, “Distance from a Point to a Plane”. Point Q and point P in-regards-to their normal vector of “n” are not given within the displayed diagram. The projection of the measured length PQ onto n, from Q and it’s distance is the result we are pursing to achieve. Here is the result we are looking for: || proj “n” PQ|| = || n(PQ) / || n ||2^ “n”|| = + n(PQ) \ || “n” || The importance provides us beyond the distance between a plane and point. Several other distances will be shown regarding the process of locating the afford mentioned. The distance from a plane, line, and parallel planes confirms proof of this main concept.The following denotes the distance from Q to h within the plane. Let Q equal the point, let the normal vector “n” of a plane be given, is denoted in the following example: h = || n(PQ) || / || n || NOTE: Any point in the plane is P. Reference: Calculus III: Gregory Hartman: 2018 APEX Publishing: Lexington, KY, 22 January 2019
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Poitier Wright - Doing my best to understand Mathematics Adding and Scaling Vector-Value Functions concept. I have also done my best to write this in my own words with the following references. Thank you for watching.
By now, hopefully our familiarity is at least somewhat proficient what it comes to understanding the real number in terms of a function output. Vector-valued functions is what we will cover here in understanding the vector’s output is a function. A specific value of “t” is straightforward when evaluating a vector-valued function. A component function at the value of “t” results in that evaluation. A prime example shows r(t) = (t2, t2 + t – 1), then r(-2) = (4, 1). This vector can be sketched although, the word “cumbersome” could come to mind when plotting a substantial amount of vectors takes fruition. Therefore, just the terminal point, versus the whole vector remains sufficient in this instance. Terminal points of r(t) is the set based on vector-valued functions in the graph. This is based upon each vector always at it’s origin, regarding it’s initial point. The graph’s indication of individual points with their respective vector is displayed as “r”. Parametric equations relevant to graphs are related closely as vector-value functions. Within the vector-valued context, functions represent a vector at each such point. To produce a graph, in both methods we plot points z(t)), y(t), (x(t) or y(t)), (x(t). The ideas of calculus to these functions are more fully realized in the next section to include the implications. Reference: Calculus III: Gregory Hartman: 2018 APEX Publishing: Lexington, KY, 22 January 2019 Hello everyone, I am doing my best to understand the following Mathematics series. Today I’ll cover the ellipsoid concept. I have also done my best to write this in my own words with the following reference As a variation of an elliptic was covered in a previous section, we’ll use the time allotted to cover this under identifying quadratic surfaces. Solutions within quadratic surface drawings, project two sheets of hyperboloi. This is derived from the equation: z2/c2 — x2/a2 — y2/b2 = 1. Along the x-axis, opens the hyperboloid, stating the only variable with a positive coefficient, must be considered from the position of: x. An equation is needed where c > b, as a wider hyperboloid is in the z-direction, opposed to the y-direction.  With a vector-valued function, where each of its component functions is constant on its domain, let t0 < t and let r(t). Ultimately, r(t) on [t0, t1] is: r(t1) – r(to) / t1 – t0 = average rate of change Here we will locate the average rate of change of r(t) with [-1, 5] and [-1, 1]. Overall, d = (0, 2) the displacement computed of r(t) on [-1, 1] is: r(1) – r(-1) / 1 - (-1) = (0, 2) / 2 = (0, 1) The following reflects the interpretation: While climbing slowly, quickly, then slowly again, proceeded by a semi-circular path object, followed. In other words, it moved quickly to the right, then back to the left. However, on average, it progressed at a constant rate straight up at (0, 1) per unit of time. The displacement on [-1, 1] is the same on [-1, 5] as we can quickly see the displacement, therefore, d = (0, 2). Although, there is a different average rate of change: r(5) – r(-1) / 5 - (-1) / (0, 2) / 6 = (0, 1/3) Reference: Calculus III: Gregory Hartman: 2018 APEX Publishing: Lexington, KY, 22 January 2019  The question arises, how can energy be conserved in the thermodynamic process which could not occur in nature? A prime example would be bringing an object into contact with a hot one. The cold object increases temperature as the hot object decreases temperature until a specific temperature is reached through equilibrium. Heat transfer of course goes from hot, to cold, objects respectively. If we visualized a system where heat was transferred from the cold object to the hot object, the first law of thermodynamics would not be violated. The conservation of energy would still take place if the hot object would get hotter, and the cold object colder. Explaining this and similar observations, since such a system in nature is not encountered, a second law of thermodynamics is proposed. Particular problems studied regarding the second law, were analyzed and researched by Carnot, Clasius, and Kelvin, This documented entails the description of the second law stated from the textbook, “Physics” by Halliday and Resnick. Entropy, a new state variable is provided in the beginning of their book. The statistical disorder of the system, are one of many physical interpretations of entropy included. Although, in covering the objective of this paper, we’ll consider that just another system’s property to be categorized as entropy such as temperature or enthalpy. Entropy exists a state that is useful based on a variable within the second law. Entropy’s change is equal to heat transfer that is divided by temperature.  Thermodynamics: First Law – Internal Energy |
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