22nd  Top calculus topics 
In mathematics, the Hessian matrix (or simply the Hessian) is the square matrix of secondorder partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse himself had used the term "functional determinants".
Given the realvalued function
if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
where x = (x_{1}, x_{2}, ..., x_{n}) and D_{i} is the differentiation operator with respect to the ith argument and the Hessian becomes
Some mathematicians ^{[1]} define the Hessian as the determinant of the above matrix.
Hessian matrices are used in largescale optimization problems within Newtontype methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
where J is the Jacobian matrix, which is a vector (the gradient) for scalarvalued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasiNewton algorithms have been developed that use approximations to the Hessian. The most wellknown quasiNewton algorithm is the BFGS algorithm.
Contents 
The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut's theorem). For example,
This can also be written as:
In a formal statement: if the second derivatives of f are all continuous in a neighborhood, D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.
If the gradient of f (i.e. its derivative in the vector sense) is zero at some point x, then f has a critical point (or stationary point) at x. The determinant of the Hessian at x is then called the discriminant. If this determinant is zero then x is called a degenerate critical point of f, this is also called a nonMorse critical point of f. Otherwise it is nondegenerate, this is called a Morse critical point of f.
The following test can be applied at a nondegenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive.
Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive (yet a conclusion can be made that f is locally convex or concave respectively). However, more can be said from the point of view of Morse theory.
In view of what has just been said, the second derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, if it is negative then x is a local maximum; if it is zero then the test is inconclusive. In two variables, the discriminant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative. If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive.
A bordered Hessian is used for the secondderivative test in certain constrained optimization problems. Given the function as before:
but adding a constraint function such that:
the bordered Hessian appears as
If there are, say, m constraints then the zero in the northwest corner is an m × m block of zeroes, and there are m border rows at the top and m border columns at the left.
The above rules of positive definite and negative definite can not apply here since a bordered Hessian can not be definite: we have z'Hz = 0 if vector z has a nonzero as its first element, followed by zeroes.
The second derivative test consists here of sign restrictions of the determinants of a certain set of n  m submatrices of the bordered Hessian.^{[2]} Intuitively, think of the m constraints as reducing the problem to one with n  m free variables. (For example, the maximization of f(x_{1},x_{2}, x_{3}) subject to the constraint x_{1} + x_{2} + x_{3} = 1 can be reduced to the maximization of f(x_{1},x_{2},1 − x_{1} − x_{2}) without constraint.)
If f is instead a function from , i.e.
then the array of second partial derivatives is not a twodimensional matrix of size , but rather a tensor of order 3. This can be thought of as a multidimensional array with dimensions , which degenerates to the previous case for m = 1.
In mathematics, the Hessian matrix (or simply the Hessian) is the square matrix of secondorder partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse himself had used the term "functional determinants".
Given the realvalued function
if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
where x = (x_{1}, x_{2}, ..., x_{n}) and D_{i} is the differentiation operator with respect to the ith argument and the Hessian becomes
\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}.
Some mathematicians ^{[1]} define the Hessian as the determinant of the above matrix.
Hessian matrices are used in largescale optimization problems within Newtontype methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
where J is the Jacobian matrix, which is a vector (the gradient) for scalarvalued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasiNewton algorithms have been developed that use approximations to the Hessian. The most wellknown quasiNewton algorithm is the BFGS algorithm.
Contents 
The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut's theorem). For example,
\frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).
This can also be written as:
In a formal statement: if the second derivatives of f are all continuous in a neighborhood D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.
If the gradient of f (i.e. its derivative in the vector sense) is zero at some point x, then f has a critical point (or stationary point) at x. The determinant of the Hessian at x is then called the discriminant. If this determinant is zero then x is called a degenerate critical point of f, this is also called a nonMorse critical point of f. Otherwise it is nondegenerate, this is called a Morse critical point of f.
The following test can be applied at a nondegenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive.
Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive (yet a conclusion can be made that f is locally convex or concave respectively). However, more can be said from the point of view of Morse theory.
In view of what has just been said, the second derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, if it is negative then x is a local maximum; if it is zero then the test is inconclusive. In two variables, the discriminant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative. If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive.
A bordered Hessian is used for the secondderivative test in certain constrained optimization problems. Given the function as before:
but adding a constraint function such that:
the bordered Hessian appears as
0 & \frac{\partial g}{\partial x_1} & \frac{\partial g}{\partial x_2} & \cdots & \frac{\partial g}{\partial x_n} \\ \\ \frac{\partial g}{\partial x_1} & \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\ \frac{\partial g}{\partial x_2} & \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial g}{\partial x_n} & \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}
If there are, say, m constraints then the zero in the northwest corner is an m × m block of zeroes, and there are m border rows at the top and m border columns at the left.
The above rules of positive definite and negative definite can not apply here since a bordered Hessian can not be definite: we have z'Hz = 0 if vector z has a nonzero as its first element, followed by zeroes.
The second derivative test consists here of sign restrictions of the determinants of a certain set of n  m submatrices of the bordered Hessian.^{[2]} Intuitively, think of the m constraints as reducing the problem to one with n  m free variables. (For example, the maximization of $f(x\_1,x\_2,x\_3)$ subject to the constraint $x\_1+x\_2+x\_3=1$ can be reduced to the maximization of $f(x\_1,x\_2,1x\_1x\_2)$ without constraint.)
If f is instead a function from $\backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{R\}^m$, i.e.
then the array of second partial derivatives is not a twodimensional matrix of size $n\; \backslash times\; n$, but rather a tensor of order 3. This can be thought of as a multidimensional array with dimensions $m\; \backslash times\; n\; \backslash times\; n$, which degenerates to the usual Hessian matrix for $m\; =\; 1$.
Let $(M,g)$ be a Riemannian manifold and $\backslash nabla$ its LeviCivita connection. Let $f:M\; \backslash to\; \backslash mathbb\{R\}$ be a smooth function. We may define the Hessian tensor $\backslash displaystyle\; \backslash mbox\{Hess\}(f)\; \backslash in\; \backslash Gamma(T^*M\; \backslash otimes\; T^*M)$ by $\backslash mbox\{Hess\}(f):=\backslash nabla\; df$. Choosing local coordinates $\backslash \{x^i\backslash \}$ we obtain the local expression for the Hessian as
where $\backslash Gamma\{ij\}^k$ are the Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by $\backslash mbox\{Hess\}(f)(X,Y)=\; \backslash langle\; \backslash nabla\_X\; \backslash mbox\{grad\}f,Y\; \backslash rangle$ and $\backslash mbox\{Hess\}(f)(X,Y)=X(Yf)(\backslash nabla\_XY)f$.
