MATH 251 Lecture 36
« previous | Monday, April 30, 2012 | next »
Written Homework 8
| FTC: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b f'(x)\,\mathrm{d}x = f(b) - f(a)} | |
| + | Product Rule: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d} x}(uv) = \left( \frac{\mathrm{d}}{\mathrm{d} x} u \right) v + u \frac{\mathrm{d}}{\mathrm{d} x} v} |
| = | Integration by Parts: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b \left(u'v\right) \,\mathrm{d}x = \left.uv\right|_a^b - \int_a^b \left(uv'\right) \,\mathrm{d}x} |
Using the divergence theorem, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint_R \mathrm{div}\vec{F} \,\mathrm{d}V = \iint_{\partial R} \vec{F} \cdot \vec{n} \,\mathrm{d}S}
Problem 1
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is a function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is a function, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}} is a vector field.
Part A
We need to verify this.
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = \left\langle P, Q, R \right\rangle} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\vec{F} = \left\langle uP, uQ, uR \right\rangle}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec\nabla\cdot \left\langle uP, uQ, uR \right\rangle &= (u_x P + u P_x) + (u_y Q + u Q_y) + (u_z R + u R_z) \\ &= (u_x P + u_y Q + u_z R) + (u P_x + u Q_y + u R_z) \\ &= (\vec\nabla u \cdot \vec{F}) + u(\vec\nabla\cdot \vec{F}) \end{align}}
Part B
Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\nabla\cdot (u \vec\nabla v)}
Plug in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = \vec\nabla v} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\nabla\cdot(u\vec\nabla v) = (\vec\nabla u) \cdot \vec\nabla v + u \Delta v} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta v} is the Laplace operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\nabla^2 v = \vec\nabla \cdot \vec\nabla v}
Problem 2
Integration by Parts:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \iiint_R \vec\nabla\cdot (u \vec{F}) \,\mathrm{d}V &= \iiint_R \left( \vec\nabla u \cdot \vec{F} + u \vec\nabla\cdot \vec{F} \right) \,\mathrm{d}V \\ \iint_{\partial R} u\vec{F} \cdot \vec{n} \,\mathrm{d}S &= \\ \iiint_R u \vec\nabla\cdot \vec{F} \,\mathrm{d}V &= \iint_{\partial R} u \vec{F} \cdot \vec{n} \,\mathrm{d}S - \iiint_R \left( \vec\nabla u \cdot \vec{F} \right) \,\mathrm{d}V \\ \end{align}}
Substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = \vec\nabla v}
:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \iiint_R u \Delta v \,\mathrm{d}V = \iint_{\partial R} u \left( \vec\nabla v \cdot \vec{n} \right) \,\mathrm{d}S - \iiint_R \left( \vec\nabla u \cdot \vec\nabla v \right) \,\mathrm{d}V \end{align}}
Problem 3
Application of this stuff: temperature at a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)i} at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
Newton's law of cooling states that heat flows against the gradient of the temperature and is proportional to the difference in temperature between two objects: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Heat\ flow} = -\alpha \vec\nabla u} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is a property of the material called heat diffusivity.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t = -\vec\nabla\cdot (-\alpha \vec\nabla u) = \alpha \Delta u}
For an insulated region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\nabla u \cdot \vec{n} = 0}
on a boundary. There is no gradient of heat leaving (pointing outward) along the boundary.
Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \iiint_R u(x,y,z,t) \,\mathrm{d}V = 0}
Since we are not integrating with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , so we can differentiate the inside:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint_R u_t \,\mathrm{d}V = 0}
Substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t = \alpha \Delta u} :
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} does not change over time, temperature enclosed must be constant.
Problem 4
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}}{\mathrm{d} t} \iiint_R \frac{1}{2} u^2 \,\mathrm{d}V &= \iiint \frac{\mathrm{d}}{\mathrm{d}t} \,\mathrm{d}V \\ &= \iiint_R u\, u_t \,\mathrm{d}V \\ &= \alpha \iiint_R u \Delta u \,\mathrm{d}V \end{align}}
The last step is from problem 3. Continue from there
Problem 6
Kinetic Energy and Potential Energy...
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(t) = \iiint_R \frac{1}{2} u_t^2 + \frac{c^2}{2} \left\| \vec\nabla u \right\|^2 \,\mathrm{d}V}
Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'(t) = 0} to show that energy is conserved.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} E'(t) &= \iiint_R u_t \, u_{tt} + c^2 \vec\nabla u \cdot \vec\nabla u_t \,\mathrm{d}V \\ &= \iiint_R u_t c^2\Delta u + c^2 \vec\nabla u \cdot \vec\nabla u_t \,\mathrm{d}V \\ &= \iint_{\partial R} u_t \vec\nabla u \cdot \vec{n} \,\mathrm{d}S \\ &= 0 \end{align}}
This is true because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = 0} along ∂R so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t = 0} as well.
Therefore energy is conserved.