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Define the function $f$

$f\left(x\,y\right)\=2 x^2\+3 y^2$$\stackrel{\text{assign as function}}{\to }$$f$

Obtain $\nabla f$ at the point $\left(1\,1\right)$ 

$\nabla f\left(x\,y\right)$ = $\stackrel{\text{evaluate at point}}{\to }$ $$

Alternate approach to obtaining $\nabla f$ at the point $\left(1\,1\right)$ 

$f\left(x\,y\right)$ = $2x^2+3y^2$$\stackrel{\text{gradient}}{\to }$$\left[\begin{array}{c}4x\\ 6y\end{array}\right]$$\stackrel{\text{evaluate at point}}{\to }$ $$

Obtain the level curve through $\left(1\,1\right)$ 

$f\left(x\,y\right)\=f\left(1\,1\right)$

$2x^2\+3y^2\=5$

$\stackrel{\text{solutions for y}}{\to }$

$\frac{1}{3}\sqrt{-6x^2\+15}\,-\frac{1}{3}\sqrt{-6x^2\+15}$

Describe the level curve as the position vector R

$⟨x\,\frac{1}{3}\sqrt{-6x^2\+15}⟩$$\stackrel{\text{assign to a name}}{\to }$$R$

Obtain a vector tangent to R at $\left(1\,1\right)$ 

$\mathbf{R}$ = $\stackrel{\text{differentiate}}{\to }$ $\stackrel{\text{evaluate at point}}{\to }$ $\stackrel{\text{simplify}}{\=}$ $\stackrel{\text{assign to a name}}{\to }$$T$$$

Verify orthogonality by the vanishing of the dot product

$\mathbf{T}·\mathbf{gf}$ = $0$$$

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$$\begin{gathered} \mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\: \\ \mathrm{BasisFormat}\left(\mathrm{false}\right)\: \end{gathered}$$

$f≔\left(x\,y\right)\to 2 x^2\+3 y^2\:$

Obtain the gradient of $f$

$\mathbf{gradf}≔\mathrm{Gradient}\left(f\left(x\,y\right)\right)$

Evaluate the gradient at $\left(x\,y\right)\=\left(1\,1\right)$ 

$\mathbf{gradf11}≔\mathrm{evalVF}\left(\mathbf{gradf}\,⟨1\,1⟩\right)$

Define the level curve through $\left(1\,1\right)$ as a position vector R

$\mathbf{ }\mathbf{R}≔⟨x\,\sqrt{\left(5-2 x^2\right)\/3}⟩$

Obtain $\mathbf{T}\left(x\right)$, a tangent vector at $x$ along the level curve

$\mathbf{T}≔\mathrm{TangentVector}\left(\mathbf{R}\right)$

Evaluate $\mathbf{T}\left(x\right)$ at $x\=1$ 

$\mathbf{T1}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathbf{T}\,x\=1\right)\right)$

Show orthogonality via the vanishing of the dot product

Apply the DotProduct command to the tangent and gradient vectors.
Because T1 is derived from a RootedVector, it retains its variable root point; the DotProduct command fails if the root points don't agree. Hence, the conversion of T1 to a free vector via the ConvertVector command.

$\mathrm{DotProduct}\left(\mathrm{ConvertVector}\left(\mathbf{T1}\,\mathrm{free}\right)\,\mathbf{gradf11}\right)$ = $0$$$