parametrized surfaces

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1: 1.6 Vectors and Vector-Valued Functions

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§1.6(v) Surfaces and Integrals over Surfaces

►A parametrized surface $S$ is defined by … ► ►The area $A(S)$ of a parametrized smooth surface is given by … ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is …

2: 21.4 Graphics

… ►Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, with …This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). ►

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Figure 21.4.1: $\widehat{\theta }\left(𝐳\right|𝛀)$ parametrized by (21.4.1). The surface plots are of $\widehat{\theta }\left(x+\mathrm{i}y,0\right|𝛀)$, $0\le x\le 1$, $0\le y\le 5$ (suffix 1); $\widehat{\theta }\left(x,y\right|𝛀)$, $0\le x\le 1$, $0\le y\le 1$ (suffix 2); $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|𝛀)$, $0\le x\le 5$, $0\le y\le 5$ (suffix 3). …

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

3: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . … ►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. … ►

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Figure 36.5.7: Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray). … Magnify

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Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify

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Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify