| Version 8 |
Version 7 |
| \PMlinkescapeword{projection} \PMlinkescapeword{flow} |
\PMlinkescapeword{projection} \PMlinkescapeword{flow} |
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| Let |
Let |
| $$\vec{U} \;=\; U_x\vec{i}+U_y\vec{j}+U_z\vec{k}$$ |
$$\vec{U} \;=\; U_x\vec{i}+U_y\vec{j}+U_z\vec{k}$$ |
| be a vector field in $\mathbb{R}^3$\, and let $a$ be a portion of some surface in the vector field.\, Define one \PMlinkescapetext{side of $a$ to be positive}; if $a$ is a closed surface, then the \PMlinkescapetext{positive side must be the outer surface} of it.\, For any surface element $da$ of $a$, the corresponding {\em vectoral surface element} is |
be a vector field in $\mathbb{R}^3$\, and let $a$ be a portion of some surface in the vector field.\, Define one \PMlinkescapetext{side of $a$ to be positive}; if $a$ is a closed surface, then the \PMlinkescapetext{positive side must be the outer surface} of it.\, For any surface element $da$ of $a$, the corresponding {\em vectoral surface element} is |
| $$d\vec{a} \;=\; \vec{n}\,da,$$ |
$$d\vec{a} \;=\; \vec{n}\,da,$$ |
| where $\vec{n}$ is the unit normal vector on the \PMlinkescapetext{positive side} of $da$. |
where $\vec{n}$ is the unit normal vector on the \PMlinkescapetext{positive side} of $da$. |
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| The {\em flux} of the vector $\vec{U}$ through the surface $a$ is the \PMlinkescapetext{surface integral} |
The {\em flux} of the vector $\vec{U}$ through the surface $a$ is the \PMlinkescapetext{surface integral} |
| $$\int_a\vec{U} \cdot d\vec{a}.$$\\ |
$$\int_a\vec{U} \cdot d\vec{a}.$$\\ |
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| \textbf{Remark.}\, One can imagine that $\vec{U}$ represents the velocity vector of a flowing liquid; suppose that the flow is \PMlinkescapetext{stationary}, i.e. the velocity $\vec{U}$ depends only on the location, not on the time.\, Then the scalar product $\vec{U} \cdot d\vec{a}$ is the volume of the liquid flown per time-unit through the surface element $da$; it is positive or negative depending on whether the flow is from the positive side to the negative \PMlinkescapetext{side} or contrarily. |
\textbf{Remark.}\, One can imagine that $\vec{U}$ represents the velocity vector of a flowing liquid; suppose that the flow is \PMlinkescapetext{stationary}, i.e. the velocity $\vec{U}$ depends only on the location, not on the time.\, Then the scalar product $\vec{U} \cdot d\vec{a}$ is the volume of the liquid flown per time-unit through the surface element $da$; it is positive or negative depending on whether the flow is from the positive side to the negative \PMlinkescapetext{side} or contrarily. |
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| \textbf{Example.}\, Let\, $\vec{U} = x\vec{i}+2y\vec{j}+3z\vec{k}$\, and $a$ be the portion of the plane \,$x+y+x = 1$\, in the first octant ($x \geqq 0,\; y \geqq 0,\, z \geqq 0$) with the \PMlinkescapetext{positive normal} away from the origin. |
\textbf{Example.}\, Let\, $\vec{U} = x\vec{i}+2y\vec{j}+3z\vec{k}$\, and $a$ be the portion of the plane \,$x+y+x = 1$\, in the first octant ($x \geqq 0,\; y \geqq 0,\, z \geqq 0$) with the \PMlinkescapetext{positive normal} away from the origin. |
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One has the constant unit normal vector:
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On has the constant unit normal vector
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| $$\vec{n} \;=\; \frac{1}{\sqrt{3}}\vec{i}+\frac{1}{\sqrt{3}}\vec{j}+\frac{1}{\sqrt{3}}\vec{k}.$$ |
$$\vec{n} \;=\; \frac{1}{\sqrt{3}}\vec{i}+\frac{1}{\sqrt{3}}\vec{j}+\frac{1}{\sqrt{3}}\vec{k}.$$ |
| The flux of $\vec{U}$ through $a$ is |
The flux of $\vec{U}$ through $a$ is |
| $$\varphi \;=\; \int_a\vec{U}\cdot d\vec{a} \;=\; \frac{1}{\sqrt{3}}\int_a(x+2y+3z)\,da.$$ |
$$\varphi \;=\; \int_a\vec{U}\cdot d\vec{a} \;=\; \frac{1}{\sqrt{3}}\int_a(x+2y+3z)\,da.$$ |
| But this surface integral may be converted to such one in which $a$ is replaced by its \PMlinkname{projection}{ProjectionOfPoint} $A$ on the $xy$-plane and $da$ similarly replaced by its projection $dA$; |
But this surface integral may be converted to such one in which $a$ is replaced by its \PMlinkname{projection}{ProjectionOfPoint} $A$ on the $xy$-plane and $da$ similarly replaced by its projection $dA$; |
| $$dA = \cos\alpha\, da$$ |
$$dA = \cos\alpha\, da$$ |
| where $\alpha$ is the angle between the normals of both surface elements, i.e. the angle between $\vec{n}$ and $\vec{k}$: |
where $\alpha$ is the angle between the normals of both surface elements, i.e. the angle between $\vec{n}$ and $\vec{k}$: |
| $$\cos\alpha \;=\; \vec{n}\cdot\vec{k} \;=\; \frac{1}{\sqrt{3}}.$$ |
$$\cos\alpha \;=\; \vec{n}\cdot\vec{k} \;=\; \frac{1}{\sqrt{3}}.$$ |
| Then we also express $z$ on $a$ with the coordinates $x$ and $y$: |
Then we also express $z$ on $a$ with the coordinates $x$ and $y$: |
| $$\varphi \;=\; \frac{1}{\sqrt{3}}\int_A(x+2y+3(1-x-y))\,\sqrt{3}\,dA |
$$\varphi \;=\; \frac{1}{\sqrt{3}}\int_A(x+2y+3(1-x-y))\,\sqrt{3}\,dA |
| \;=\; \int_0^1\left(\int_0^{1-x}(3-2x-y)\,dy\right)dx \;=\; 1$$ |
\;=\; \int_0^1\left(\int_0^{1-x}(3-2x-y)\,dy\right)dx \;=\; 1$$ |
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