MATH 251 Lecture 12

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Differential = Derivative = best linear approximation

Directional Derivative

Given 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 f \left( x,\, y \right)} and a vector ${\vec {v}}$ (assuming 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{v}} is a unit vector?).

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 \left(D_\vec{v} f\right)\left(x,\,y\right)} is the derivative of 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 f} in the direction of 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{v}} .

The path is 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 \left( x + v_x t ,\, y + v_y t \right)}

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} D_\vec{v} f\left(x,\,y\right) &= \frac{\mathrm{d}}{\mathrm{d}t}\, f\left( x+v_x t ,\, y+v_y t \right) \\ &= \frac{\partial f}{\partial x} \left( \frac{\mathrm{d}}{\mathrm{d} t} \left( x + v_x t \right) \right) + \frac{\partial f}{\partial y} \left( \frac{\mathrm{d}}{\mathrm{d} t} \left( y + v_y t \right) \right) \\ &= \mathrm{d}f \left( v_x,\, v_y \right) \\ &= \frac{\partial f}{\partial x} \left( v_x \right) + \frac{\partial f}{\partial y} \left( v_y \right) \\ &= \vec\nabla f \cdot \hat{v} \end{align}}

Example

Find the directional derivative of 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^2y + y\sin{x}} in the direction of 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{v} = \left\langle 1,\, 1 \right\rangle} at 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 \left( \pi,\, 2 \right)} .

Answer 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 \begin{align} \left. \frac{\partial f}{\partial x} \right|_{\left(\pi,2\right)} &= \left. 2xy+y\cos{x} \right|_{\left(\pi,2\right)} \\ &= 4\pi - 2 \\ \left. \frac{\partial f}{\partial y} \right|_{\left(\pi,2\right)} &= \left. x^2+\sin{x} \right|_{\left(\pi,2\right)} \\ &= \pi^2 \\ \left( 4\pi - 2 \right) \left( 1 \right) + \left( \pi^2 \right) \left( 1 \right) &= 4\pi - 2 + \pi^2 \end{align}}

Answer 2

Convention: Direction always means unit direction. Replace 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{v}} with 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 \hat{v}} . This is the way that the book uses.

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} \hat{v} &= \frac{\vec{v}}{\left\| \vec{v} \right\|} = \frac{1}{\sqrt{2}} \\ \left( 4\pi-2 \right) \left( \frac{1}{\sqrt{2}} \right) + \frac{\pi^2}{\sqrt{2}} &= \frac{4\pi-2+\pi^2}{\sqrt{2}} \end{align}}

Gradient

A vector quantity

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 \nabla f \left( x,\, y \right) = \left\langle \frac{\partial f}{\partial x},\, \frac{\partial f}{\partial y} \right\rangle}

Valid in euclidean rectangular coordinates.

Facts:

1. It ponts in the direction of steepest ascent of 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 f} .
2. Points orthogonally to the level curves of 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 f} .

Example

Find the gradient of 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 f(x,y) = \sin^2{x} \mathrm{e}^y}

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} \nabla f (x,y) &= 2\sin{x}\cos{x} \mathrm{e}^y \hat\imath + \sin^2{x}\mathrm{e}^y \hat\jmath \\ \mbox{at } \left( \frac{3\pi}{2},\, 1 \right) &= \left\langle 0,\, \mathrm{e} \right\rangle \\ \mbox{Rate of Change} = \left\| \nabla f \right\| &= \sqrt{0^2 + \mathrm{e}^2} = \mathrm{e} \end{align}}