Reference Pages
Bioengineering

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    Conduction

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    Steady State

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    Unsteady State

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    Convection

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    Heat Transfer with Phase Change

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    Radiative Heat Transfer

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    Governing Equations & Boundary Conditions of Heat Transfer

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    General Heat Transfer Equation

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    Boundary Conditions

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    Table Example

    Table Example

    abc
    def
    ghi
    jkl

    Formula Box

    Temperature unit conversions
    $$ \begin{aligned}\textrm{Fahrenheit to Celcius:}&~ ^{\circ}\textrm{C} = (^{\circ}\textrm{F} -32)X (9/5) \\ \textrm{Celsius to Kelvin:}&~ ^{\circ}\textrm{K} = ^{\circ}\textrm{C} + 273.15\end{aligned} $$

    Formula Box With Derivation Drop Down

    Pythagorus' length formula. #rvv-ey
    $$ a = \|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} $$

    First we prove Pythagorus' theorem for right-angle triangles. For side lengths \(a\) and \(b\) and hypotenuse \(c\), the fact that \(a^2 + b^2 = c^2\) can be seen graphically below, where the gray area is the same before and after the triangles are rotated in the animation:

    Pythagorus' theorem immediately gives us vector lengths in 2D. To find the length of a vector in 3D we can use Pythagorus' theorem twice, as shown below. This gives the two right-triangle calculations:

    $$ \begin{aligned} \ell^2 &= a_1^2 + a_2^2 \\ a^2 &= \ell^2 + a_3^2 = a_1^2 + a_2^2 + a_3^2. \end{aligned} $$

    Click and drag to rotate.
    Warning: Length must be computed in a single basis. #rvv-wl
    The Pythagorean length formula can only be used if all the components are written in a single orthonormal basis.

    Image

    Test Image: Lecture Notes

    Example Problem

    Example Problem: Example #undefined

    Example

    $$ 2 $$

    solution explanation

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