MA35 Labs

MA35 – Multivariable Calculus Labs

Number of Variables: [1] [2] [3] [n]

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Contents: 2 Variables

2.1 Functions of Two Variables

2.2 Differentiation

2.3 More Derivatives

2.3.1 Directional Derivative

2.3.2 Second Partial Derivatives

2.3.3 Hessian Determinant

2.3.4 Taylor Series

2.3.5 Constrained Maxima & Minima

2.3.6 Equality of Mixed Partials

2.4 Integration

2.5 Vector Analysis

Directional Derivative

The directional derivative for a function f (x,y) in the direction θ is denoted ∇θf (x,y) and is the derivative of the height function z(t) in that direction.

The directional derivative ∇θf (x0,y0) at the point P will be the derivative evaluated at t = 0 of the height fuction z(t)

\left. \nabla_{\theta}f(x_0,y_0) = z'(t) \right|_{t=0} = \frac{\partial}{\partial t} \left. f(x_{0}+t\cos\theta,y_{0}+t\sin\theta) \right|_{t=0}.

Demos

Tangent Line to Slice Curves

Examples

Exercises

Find the value of θ for the maximum directional derivative of any slice curve. What θ gives the minimum directional derivative for this same slice curve? How are these values related? Try this for several different slice curves.

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