Concept: Derivatives can be found while holding one or more variables constant (editing)

Prerequisite concepts
Differential Calculus
Variables can be held constant

When taking a derivative of a multivariable function, it is possible to consider a simpler situation in which some of the variables are considered to be constant.

The Heater II

Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).

Representations used

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Leibniz Notation

$\left(\frac{\partial f}{\partial x}\right)_y$

Using Leibniz notation, variables that are held constant when taking a partial derivative are denoted with subscripts
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Contour Maps

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Surface

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The Hot Plate
The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.
There is a partial derivative in every direction at any point
There is a tangent line in every direction at every point
The magnitude of the gradient is proportional to the density of contour lines
Partial Derivative Machine Derivatives
Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

FIXME
There are experimental limits to how $small$ of a change can be measured
The derivative can be interpreted physically
The coefficients in a differentials equation are partial derivatives
The Hillside
The direction of the gradient is perpendicular to contour lines
The direction of the gradient shows what way is 'uphill'
The magnitude of the gradient is proportional to the density of contour lines
The components of the gradient are partial derivatives
The Hill
The magnitude of the gradient is proportional to the density of contour lines
The direction of the gradient is perpendicular to contour lines
The gradient is a local quantity
The gradient lives in the domain and has the same number of spatial dimensions as the original function
Chain Rules
There might be experimental limits on which quantities you can measure
Differentials allow the finding of partial derivatives when a variable cannot be solved for
You can flip a partial derivative if the same variable(s) are constant
Guitar String Lecture (8)
Developing physical interpretations of partial derivatives (dimensions)
Second partial Derivatives