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1: 19.18 Derivatives and Differential Equations

… ►Let ${\partial }_j=\partial /\partial z_j$, and ${\mathbf{e}}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … ►If $n=2$, then elimination of ${\partial }_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_1,b_2,z_1,z_2)=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). … ►

19.18.14 $$\frac{{\partial }^{2}w}{{\partial x}^2}=\frac{{\partial }^{2}w}{{\partial y}^2}+\frac{1}{y}\frac{\partial w}{\partial y}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/19.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.15 $$\frac{{\partial }^{2}W}{{\partial t}^2}=\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential and $W$: wave function

Permalink:

http://dlmf.nist.gov/19.18.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.16 $$\frac{{\partial }^{2}u}{{\partial x}^2}+\frac{{\partial }^{2}u}{{\partial y}^2}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/19.18.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

…

2: 36.10 Differential Equations

… ►

§36.10(ii) Partial Derivatives with Respect to the $x_n$

… ►

36.10.7 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_2}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_2}{{\partial y}^n}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

… ►

36.10.8 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial y}^n},$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

… ►

36.10.10 $$\frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial y}^{3n}}={\mathrm{i}}^n\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial z}^{2n}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $y$: real parameter, $z$: real parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/36.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

… ►

§36.10(iv) Partial $z$-Derivatives

…

3: 16.14 Partial Differential Equations

§16.14 Partial Differential Equations

►

§16.14(i) Appell Functions

►

$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

… ►

$$x(1-x)\frac{{\partial }^2{F}_2}{{\partial x}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_2}{\partial x}-\beta y\frac{\partial F_2}{\partial y}-\alpha \beta F_2=0,$$

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_{2}F_1$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

4: 10.38 Derivatives with Respect to Order

… ►

10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

►

10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). ►

10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

… ►

${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=0.$

…

5: 1.5 Calculus of Two or More Variables

… ►

§1.5(i) Partial Derivatives

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime }(x)=-F_x/F_y$. … ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. … ►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. …

6: 10.15 Derivatives with Respect to Order

… ►

10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

… ►

10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►For $\partial J_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.2.4) and (10.15.3). … ►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

…

7: 10.73 Physical Applications

… ►

10.73.1 $${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{{\partial \varphi }^2}+\frac{{\partial }^{2}V}{{\partial z}^2}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.3 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\sin z$: sine function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

…

8: 12.17 Physical Applications

… ►

12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►

12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

…

9: 28.32 Mathematical Applications

… ►

28.32.2 $$\frac{{\partial }^{2}V}{{\partial x}^2}+\frac{{\partial }^{2}V}{{\partial y}^2}+k^{2}V=0$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $x$: real variable, $y$: real variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

… ►

28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

… ►

28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

… ►

28.32.5 $$K(z,\zeta )\frac{du(\zeta )}{d\zeta }-u(\zeta )\frac{\partial K(z,\zeta )}{\partial \zeta }$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable, $\zeta$: variable, $u(\zeta )$: solution and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

…

10: 1.6 Vectors and Vector-Valued Functions

… ►

1.6.20 $$gradf=\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}.$$

ⓘ

Defines:

$grad$: gradient of differentiable scalar function

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: variable, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

1.6.21 $$div\mathbf{F}=\nabla \cdot \mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$$

ⓘ

Defines:

$div$: divergence of vector-valued function

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: variable and $F_j$: vector function components

Permalink:

http://dlmf.nist.gov/1.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

1.6.46 $${\mathbf{T}}_u=\frac{\partial x}{\partial u}(u_0,v_0)\mathbf{i}+\frac{\partial y}{\partial u}(u_0,v_0)\mathbf{j}+\frac{\partial z}{\partial u}(u_0,v_0)\mathbf{k}$$

ⓘ

Symbols:

$(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\mathbf{i}$: unit vector, $\mathbf{j}$: unit vector and $\mathbf{k}$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

… ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … ►where $\partial g/\partial n=\nabla g\cdot \mathbf{n}$ is the derivative of $g$ normal to the surface outwards from $V$ and $\mathbf{n}$ is the unit outer normal vector. …