Multivariable Analysis Notes

Multivariable Analysis

Clairaut’s theorem

Suppose that two mixed \(n\)-th ordered partial derivatives of a function involve the same differentiations, but in different orders. If those partial derivatives are continuous at a point , and if and all partial derivatives of of order less than are continuous in a neighbourhood of \(p\), then the two mixed \(n\)-th ordered partial derivatives of are equal.

Directional derivatives

Let \(n \geq 2\) be a natural number, let \(R\) be a non-empty open region of \(\mathbb{R}^{n}\), let \(x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in R\) and let \(\mathbf{u}\) be a given vector in \(\mathbb{R}^{n}\). Denoting the unit vector in the direction of \(\mathbf{u}\) by \(\widehat{\mathbf{u}}=\left\langle u_{1}, u_{2}, \ldots, u_{n}\right\rangle\) and letting \(f: R \rightarrow \mathbb{R}\), the derivative of \(f\) at \(x\) the direction of \(u\), is the limit

\[D_{\mathbf{u}} f(x):=\lim _{h \rightarrow 0} \frac{f\left(x_{1}+h u_{1}, \ldots, x_{n}+h u_{n}\right)-f\left(x_{1}, \ldots, x_{n}\right)}{h} \]

provided that the limit exist.

Gradient vector

Assume the setting of Proposition \(3.1\) and that the partial derivative of \(f\) with respect to \(x_{i}\) at \(a\) exists, for all \(i \in\{1,2, \ldots, n\}\). The gradient vector of \(f\) at \(a\) is the vector

\[\nabla f(a):=\frac{\partial}{\partial x_{1}} f(a) \mathbf{x}_{1}+\frac{\partial}{\partial x_{2}} f(a) \mathbf{x}_{2}+\cdots+\frac{\partial}{\partial x_{n}} f(a) \mathbf{x}_{\mathbf{n}} \]

where \(\mathbf{x}_{1}=\langle 1,0,0 \ldots, 0\rangle, \mathbf{x}_{2}=\langle 0,1,0 \ldots, 0\rangle, \ldots, \mathbf{x}_{n}=\langle 0,0, \ldots, 0,1\rangle\) denote the standard unit vectors of \(\mathbb{R}^{n}\).

Remark

• $\nabla f(p) = 0 $ infers that \(p\) is a stationary point
• $\nabla f(p) $ Doesn't exist infers that \(p\) is a stationary point.

A recap from linear algebra

If \(A=\left(a_{i, j}\right)_{i, j=1}^{n}\), then the leading minors of \(A\) are the values

\[\operatorname{det}\left(A_{1}\right)=\left|a_{11}\right|, \quad \operatorname{det}\left(A_{2}\right)=\left|\begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array}\right|, \quad \ldots, \quad \operatorname{det}\left(A_{n}\right)=\left|\begin{array}{cccc} a_{1,1} & a_{1,2} & \ldots & a_{1, n} \\ a_{2,1} & a_{2,2} & \ldots & a_{2,3} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n, 1} & a_{n, 2} & \ldots & a_{n, n} \end{array}\right| \]

Recall that if \(A\) is an \(n \times n\) matrix, letting

\[\mathbf{a}_{\mathbf{1}}=A\left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right), \quad \mathbf{a}_{\mathbf{2}}=A\left(\begin{array}{c} 0 \\ 1 \\ \vdots \\ 0 \end{array}\right), \quad \ldots, \quad \mathbf{a}_{\mathbf{n}}=A\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \end{array}\right) \]

then \(A\) maps the unit \(n\)-dimensional cube to the \(n\)-dimensional parallelotope \(P\) defined by the vectors a \(_{1}\), a \(_{2}\), \(\ldots, \mathbf{a}_{\mathbf{n}}\), namely the region \(P=\left\{c_{1} \mathbf{a}_{\mathbf{1}}+\cdots+c_{n} \mathbf{a}_{\mathbf{n}}: 0 \leq c_{i} \leq 1\right.\) for all \(\left.i \in\{1,2, \ldots n\}\right\} .\) The determinant gives the signed \(n\)-dimensional volume of \(P\), that is, \(\operatorname{det}(A)=\pm \operatorname{vol}(P)\), and hence \(\operatorname{describes} n\)-dimensional volume scaling factor of the linear transformation \(A\). The sign shows whether the transformation preserves or reverses orientation. In particular, if the determinant is zero, then \(P\) has zero \(n\)-dimensional volume and thus the dimension of the image of \(A\) is strictly less than \(n\). This means that \(A\) produces a linear transformation which is not injective, and so is not invertible.

A symmetric \(n \times n\) matrix \(A\) is said to be positive definite if the scalar \(z^{t} A z\) is strictly positive for every non-zero column vector z.

A symmetric \(n \times n\) matrix \(A\) is said to be negative definite if the scalar \(z^{t} A z\) is strictly negative for every non-zero column vector \(\mathbf{z}\).

Indeed, if \(A\) is a positive or negative definite matrix, then \(\operatorname{det}(A) \neq 0\). Further, \(A\) is positive definite if and only if \(\operatorname{det}\left(A_{r}\right)>0\) for all \(r \in\{1,2, \ldots, n\} ; A\) is negative definite if and only if \((-1)^{r} \operatorname{det}\left(A_{r}\right)>0\) for all \(r \in\{1,2, \ldots, n\} .\) Another equivalent definition of \(A\) being positive definite, is to say that all of the eigenvalues of \(A\) are positive; and thus an equivalent definition of \(A\) being negative definite, is to say that all of the eigenvalues of \(A\) are negative.

The leading minor test

Let \(D \subset \mathbb{R}^{2}\) have non-empty interior. Suppose \(f: D \rightarrow \mathbb{R}\) is such that all its second order partial derivatives are continuous at a point \((a, b) \in D\), ant that \((a, b)\) is a stationary point of \(f .\) If \(H=H(a, b)\) denotes the Hessian of \(f\) at \((a, b)\) and if \(\operatorname{det}(H) \neq 0\), then \((a, b)\) is

• a local maximum if \(\operatorname{det}\left(H_{1}\right)<0\) and \(\operatorname{det}(H)>0\)😭\(H\) is negative definite)
• a local minimum if \(\operatorname{det}\left(H_{1}\right)>0\) and \(\operatorname{det}(H)>0\)😭\(H\) is positive definite)
• a saddle point if neither of the above hold.
  Let \(D \subset \mathbb{R}^{3}\) have non-empty interior. Suppose \(f: D \rightarrow \mathbb{R}\) is such that all its second order partial derivatives are continuous at a point \((a, b, c) \in D\), and that \((a, b, c)\) is a stationary point of \(f\). If \(H=H(a, b, c)\) denote the Hessian of \(f\) at \((a, b, c)\) and if \(\operatorname{det}(H) \neq 0\), then \((a, b, c)\) is
• a local maximum if \(\operatorname{det}\left(H_{1}\right)<0, \operatorname{det}\left(H_{2}\right)>0\) and \(\operatorname{det}\left(H_{3}\right)<0\); (\(H\) is negative definite)
• a local minimum if \(\operatorname{det}\left(H_{1}\right)>0, \operatorname{det}\left(H_{2}\right)>0\) and \(\operatorname{det}\left(H_{3}\right)>0\)😭\(H\) is positive definite)
• a saddle point if neither of the above hold.
  In each of the above cases, if \(\operatorname{det}(H)=0\), then \(p\) can be either a local extremum or a saddle point

Remark

• \(H\) is positive definite infers that \(p\) is a local minimum;
• \(H\) is negative definite infers that \(p\) is a local maximum;
• $H \neq 0 $ Infers that \(p\) is a saddle point;
• \(H = 0\) infers nothing!!

Taylor Series - Function of multivariables

Suppose that all partial derivatives of all order of a function \(f\) of two variables exist and are continuous on an open connected domain \(I \subseteq \mathbb{R}^{2} .\) Let \((a, b)\) denote a point in \(I\) and let \(h=\left(h_{1}, h_{2}\right) \in \mathbb{R}^{2}\) be such that \((a, b)+t\left(h_{1}, h_{2}\right)=\left(a+t h_{1}, b+h_{2}\right) \in I\) for all \(t \in[-1,1]\). The Taylor series generated by \(f\) centred at \((a, b)\) and evaluated at \(\left(a+h_{1}, b+h_{2}\right)\) is the power series

\[\begin{aligned} &\sum_{j=0}^{\infty} \frac{(\mathbf{h} \cdot \nabla)^{j} f(a, b)}{j !} \\ &=f(a, b)+h_{1} f_{x}(a, b)+h_{2} f_{y}(a, b) \\ &\quad+\frac{1}{2 !}\left(h_{1}^{2} f_{x x}(a, b)+2 h_{1} h_{2} f_{x y}(a, b)+h_{2}^{2} f_{y y}(a, b)\right) \\ &\quad+\frac{1}{3 !}\left(h_{1}^{3} f_{x x x}(a, b)+3 h_{1}^{2} h_{2} f_{x x y}(a, b)+3 h_{1} h_{2}^{2} f_{x y y}(a, b)+h_{2}^{3} f_{y y y}(a, b)\right)+\ldots, \end{aligned} \]

where \(\mathbf{h}=\left\langle h_{1}, h_{2}\right\rangle\) is the vector with initial point at the origin and terminal point at \(h\).

Remark

Fubini-Tonelli's Theorem

Let \(D\) denote a region of \(\mathbb{R}^{2}\) and let \(f\) denote a real-valued function of two variables \(x\) and \(y\), with domain \(D\). If one of

\[\iint_{D}|f(x, y)| \mathrm{d} A, \quad \iint_{D}(|f(x, y)| \mathrm{d} x) \mathrm{d} y, \quad \iint_{D}(|f(x, y)| \mathrm{d} y) \mathrm{d} x \]

exist, then

\[\iint_{D} f(x, y) \mathrm{d} A=\iint_{D}(f(x, y) \mathrm{d} x) \mathrm{d} y=\iint_{D}(f(x, y) \mathrm{d} y) \mathrm{d} x . \]

In other words, the double integral of \(f\) over \(D\) is equal to a repeated integral where we first integrate with respect to \(x\) and then with respect to \(y\), which in turn is equal to the repeated integral where we first integrate with respect to \(y\) and then with respect to \(x\).

Remark

Fubini's Theorem states that, if the integral exists, then the order of integral can change arbitrarily.

Co-ordinate systems

• Cylindrical polar co-ordinates
• Spherical polar co-ordinates

Change of Variables

Cylindrical polar co-ordinates

\[\mathrm{d}x \mathrm{d}y = r \mathrm{d}r \mathrm{d} \theta \]

Spherical polar co-ordinattes

\[\iiint_{E} f(x, y, z) d V\quad=\int_{c}^{d} \int_{\alpha}^{\beta} \int_{a}^{b} f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi) \rho^{2} \sin \phi d \rho d \theta d \phi \]

where \(E\) is a spherical wedge given by

\[E=\{(\rho, \theta, \phi) \mid \mathrm{a} \leqslant \rho \leqslant b, \alpha \leqslant \theta \leqslant \beta, c \leqslant \phi \leqslant d\} \]

Jacobian and Change of Variables

Suppose that \(x\) and \(y\) are functions of two variables \(u\) and \(v\), whose partial derivatives exist, then the Jacobian of \(x=x(u, v)\) and \(y=y(u, v)\) with respect to \(u\) and \(v\) is the determinant

\[\frac{\partial(x, y)}{\partial(u, v)}:=\left|\begin{array}{ll} x_{u} & x_{v} \\ y_{u} & y_{v} \end{array}\right| \]

Similarly, suppose that \(x, y\) and \(z\) are functions of three variables \(u, v\) and \(w\), whose partial derivatives exist, then the Jacobian of \(x=x(u, v, w), y=y(u, v, w)\) and \(z=z(u, v, w)\) with respect to \(u, v\) and \(w\) is the determinant

\[\frac{\partial(x, y, z)}{\partial(u, v, w)}:=\left|\begin{array}{lll} x_{u} & x_{v} & x_{w} \\ y_{u} & y_{v} & y_{w} \\ z_{u} & z_{v} & z_{w} \end{array}\right| . \]

\[\mathrm{d} x \mathrm{~d} y \mathrm{~d} z=\mathrm{d} V=\left|\frac{\partial(x, y, z)}{\partial(u, v, w)}\right| \mathrm{d} u \mathrm{~d} v \mathrm{~d} w \]

Div and Curl

Definition is omitted.

If \(\mathbf{u}\) and \(\mathbf{v}\) are \(C^{2}\)-vector fields, and if \(w\) is a scalar field whose first and second order partial derivatives are continuous, then the following equalities hold.

• \(\nabla \cdot(\mathbf{u}+\mathbf{v})=\nabla \cdot \mathbf{u}+\nabla \cdot \mathbf{v}\) or equivalently \(\operatorname{div}(\mathbf{u}+\mathbf{v})=\operatorname{div}(\mathbf{u})+\operatorname{div}(\mathbf{v})\)

• \(\nabla \times(\mathbf{u}+\mathbf{v})=\nabla \times \mathbf{u}+\nabla \times \mathbf{v}\) or equivalently \(\operatorname{curl}(\mathbf{u}+\mathbf{v})=\operatorname{curl}(\mathbf{u})+\operatorname{curl}(\mathbf{v})\)

• \(\nabla \cdot(w \mathbf{u})=\nabla(w) \cdot \mathbf{u}+w \nabla \cdot \mathbf{u}\)

• \(\nabla \times(w \mathbf{u})=\nabla w \times \mathbf{u}+w \nabla \times \mathbf{u}\)

• \(\operatorname{curl}(\nabla(w))=\nabla \times \nabla(w)=\mathbf{0}\)

• \(\operatorname{div}(\operatorname{curl}(\mathbf{u}))=\nabla \cdot(\nabla \times \mathbf{u})=0\)

• \(\operatorname{div}(\nabla(w))=\Delta(w)\), where \(\Delta\) denotes the Laplacian

Fundamental Theorem of calculus for line integral

Let \(a\) and \(b \in \mathbb{R}\) with \(a<b\), let \(n \geq 2\) denote a natural number, let \(\Omega\) denote a connected non-empty open subset of \(\mathbb{R}^{n}\), and let \(f: \Omega \rightarrow \mathbb{R}\) be a scalar field of the variables \(x_{1}, x_{2}, \ldots, x_{n}\), such that all of its partial derivatives are continuous. If \(\mathbf{r}:[a, b] \rightarrow \mathbb{R}^{n}\) is a piecewise \(C^{1}\) parametrisation of a curve \(C\) contained in \(\Omega\), then

\[\int_{C} \nabla(f) \cdot \mathrm{d} \mathbf{r}=f(\mathbf{r}(b))-f(\mathbf{r}(a)) . \]

Remark

• \(\bold r\) is piecewise \(C^1\) parametrisation
• all partial derivatives of \(f\) are continuous

Green's Theorem

\[\oint_{C} \mathbf{F} \cdot \mathrm{d} \mathbf{r}=\oint_{C} F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y=\iint_{R}\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\right) \mathrm{d} A \]

Remark

• \(R\) is regular closed region
• \(\partial R = C\) consist of one or more piecewise \(C^1\) simple closed curves
• \(C\) positive oriented with respect to \(R\)
• \(\bold r\) is a piecewise \(C^1\) parameterisation of \(C\)
• \(r\) traverse \(C\) once with alignment with the orientation induced by \(R\)

Remark

Green's theorem is a special case of Stoke's theorem, but it requires a relatively relaxed conditions.

Stokes' Theorem

\[\oint_{C} \mathbf{F} \cdot \mathrm{d} \mathbf{r}=\iint_{R} \operatorname{curl}(\mathbf{F}) \cdot \mathbf{n} \mathrm{d} S = \iint_{R} \operatorname{curl}(\mathbf{F}) \cdot \mathrm{d}\mathbf{S} \]

Remark

• \(R\) oriented, piecewise smooth
• \(C\) simple, closed, piecewise smooth, positive orientation
• \(\bold F\) smooth and contains \(S\)

Gauss' Divergence Theorem

\[\underbrace{\iint_{S} \mathbf{F} \cdot \mathbf{n} \mathrm{d} S}_{\begin{array}{c} \text { Net outward flux } \\ \text { of } \mathrm{F} \text { across } S \end{array}}=\underbrace{\iiint_{V} \operatorname{div}(\mathbf{F}) \mathrm{d} V}_{\begin{array}{c} \text { Divergence } \\ \text { integral } \end{array}} \]

• \(V\) non-empty closed solid in \(\mathbb{R}^3\)
• \(\partial V = S\) smooth, orientable, closed
• \(\bold n\) with positive (outward) of \(V\)
• \(\bold F\) smooth