Concept: There is a partial derivative in every direction at any point

Prerequisite concepts

Differential Calculus "With respect to what" matters

Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".

The Heater II The Hot Plate Chain Rule Measurements

There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.

Just as the derivative of a one-dimensional function at a point is equal to the slope of a tangent line at that point, and partial derivative of a multivariable function can be found as a slope of a tangent plane.

The components of the gradient 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

There is a partial derivative in every direction at any point

Representations used

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