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# 物理代写|结构力学代写Structural Mechanics代考|CIVL2330 Scalar fields of vector variables

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## 物理代写|结构力学代写Structural Mechanics代考|Scalar fields of vector variables

Scalar fields of vector variables. A scalar field is a function $g(\mathbf{x})$ that assigns a scalar value to each point $\mathbf{x}$ in a particular domain. The temperature in a solid body is an example of a scalar field. As an example consider the scalar field $g(\mathbf{x})=|\mathbf{x}|^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$, in which the function $g(\mathbf{x})$ gives the square of the length of the position vector $\mathbf{x}$. In two dimensions, a scalar field can be represented by either a graph or a contour map like those shown in Fig. $18 .$
As with any function that varies from point to point in a domain, we can ask the question: At what rate does the field change as we move from one point to another? It is fairly obvious from the contour map that if one moves from one point to another along a contour then the change in the value of the function is zero (and therefore the rate of change is zero). If one crosses contours then the function value changes. Clearly, the question of rate of change depends upon direction of the line connecting the two points in question.

Consider a scalar field $g$ in three dimensional space evaluated at two points $a$ and $b$, as shown in Fig. 19. Point $a$ is located at position $\mathbf{x}$ and point $b$ is located at position $\mathbf{x}+\Delta s \mathbf{n}$, where $\mathbf{n}$ is a unit vector that points in the direction from $a$ to $b$ and $\Delta s$ is the distance between them. The directional derivative of the function $g$ in the direction $\mathbf{n}$, denoted $D g \cdot \mathbf{n}$, is the ratio of the difference in the function values at $a$ and $b$ to the distance between the points, as the point $b$ is taken closer and closer to $a$

$$D g(\mathbf{x}) \cdot \mathbf{n} \equiv \lim {\Delta s \rightarrow 0} \frac{g(\mathbf{x}+\Delta s \mathbf{n})-g(\mathbf{x})}{\Delta s}$$ The directional derivative of $g$ can be computed, using the chain rule of differentiation, from the formula $$D g(\mathbf{x}) \cdot \mathbf{n}=\frac{d}{d \varepsilon}(g(\mathbf{x}+\varepsilon \mathbf{n})){\varepsilon=0}=\frac{\partial g}{\partial x_{i}} n_{i}$$
In essence, the directional derivative determines the one-dimensional rate of change (i.e., $d / d \varepsilon$ ) of the function at the point $\mathbf{x}$ and just starting to move in the fixed direction $\mathbf{n}$. Because $\mathbf{x}$ and $\mathbf{n}$ are fixed, the derivative is an ordinary one.

## 物理代写|结构力学代写Structural Mechanics代考|Vector fieldS

Vector fields. A vector field is a function $\mathbf{v}(\mathbf{x})$ that assigns a vector to each point $\mathbf{x}$ in a particular domain. The displacement of a body is a vector field. Each point of the body moves by some amount in some direction. The force induced by gravitational attraction is a vector field.

Figure 21 shows two examples of vector fields. The pictures show the vectors at only enough points to get the idea of how the vectors are oriented and sized. The second vector field shown in the figure can be expressed in functional form as
$$\mathbf{v}(\mathbf{x})=x_{1} \mathbf{e}{1}+x{2} \mathbf{e}_{2}$$

The vectors point in the radial direction, and their length is equal to the distance of the point of action to the origin.

In general, if our base vectors are assumed to be constant throughout our domain, then the vector field can be expressed in terms of component functions
$$\mathbf{v}(\mathbf{x})=v_{i}(\mathbf{x}) \mathbf{e}{i}$$ For example, from Eqn. (76) we can see that the explicit expression for the components of the vector field are $v{1}(\mathbf{x})=x_{1}, v_{2}(\mathbf{x})=x_{2}$, and $v_{3}(\mathbf{x})=0$. For curvilinear coordinates, the base vectors are also functions of the coordinates.

## 物理代写|结构力学代写STRUCTURAL MECHANICS代 考|SCALAR FIELDS OF VECTOR VARIABLES

$$D g(\mathbf{x}) \cdot \mathbf{n} \equiv \lim \Delta s \rightarrow 0 \frac{g(\mathbf{x}+\Delta s \mathbf{n})-g(\mathbf{x})}{\Delta s}$$

$$D g(\mathbf{x}) \cdot \mathbf{n}=\frac{d}{d \varepsilon}(g(\mathbf{x}+\varepsilon \mathbf{n})) \varepsilon=0=\frac{\partial g}{\partial x_{i}} n_{i}$$

\mathbf{v}(\mathbf{x})=x_{1} \mathbf{e} 1+x 2 \mathbf{e}{2} $$向量指向径向，它们的长度等于作用点到原点的距离。 一般来说，如果假设我们的基向量在整个域中是恒定的，那么向量场可以用分量函数表示$$ \mathbf{v}(\mathbf{x})=v{i}(\mathbf{x}) \mathbf{e} i


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