2.6. Gradient of the function.

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Del curso dictado por National Research University Higher School of Economics

Mathematics for economists

9 calificaciones

National Research University Higher School of Economics

Mathematics for economists

9 calificaciones

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

De la lección

Differentiation. Gradient. Hessian.

Week 2 of the Course is devoted to the main concepts of differentiation, gradient and Hessian. Of special attention is the chain rule. Also students will understand economic applications of the gradient.

2.1. Differentiation. Partial differentiation.7:38

2.2. Example of differentiation. Cobb-Douglas function.9:32

2.3. Tangent plane.7:46

2.4. Total differential.7:05

2.5. Chain rule for multivariate functions.8:05

2.6. Gradient of the function.9:58

2.7. Economic applications of the gradient.9:31

2.8. Equation of a circumference. Smooth curves.13:42

2.9. Chain rule for differentiation.10:07

2.10. Linear approximation. Example of tangent plane for particular function.10:40

2.11. Second-order derivatives.10:34

2.12. Young's Theorem.5:20

2.13. Hessian matrix.5:24

Conoce a los instructores

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

Back to the straight line,

this parametric equation can be written in a vector form like this.

Now, what happens if we take the function either N dimensional space

and substitute X,

which is a point belonging to the straight line inside the function.

So, we replace X with the X which is a point in the straight line.

So, then Y equals F and here we have

zero plus LT. Now,

this is a fixed point.

L is a direction on the line and T is the venerable.

Now, we're introducing the rate of change

of the function in direction L.

The definition is as follows,

this rate of change is given by the derivative.

So, we take derivative of this composite function

because we have inserted values.

This is a vector which belongs to the straight line.

So, we're considering an ordinary derivative with respect

to T. Now,

at what point this derivative should be taken at

t equals zero or rather at the point X zero.

Now, we will try to find out how to calculate this ordinary derivative.

In order to do that,

we'll finish the statement here.

But firstly, let me derive the expression we're looking for.

We'll base our calculation on total differential.

So, total differentials we know is df represented.

This time we're using the symbol of summation,

because we're dealing with a N dimensional space.

Here each partial derivative with respect to corresponding coordinate XI,

is multiplied by the differential of the independent variable.

So, if we take the total differential N,

divide the left hand side and the right hand side by

the increment in parameter T, what do we get then?

Left hand side, DF over DT,

this is exactly the rate we're looking for.

Here will have.

But don't forget we have the parametric equation of the straight line.

So, when we take the Ith coordinate or

this equation and differentiate it with respect to T,

quite often T is regarded as time.

Then, what do we get from this formula?

What we get is

the Ith component or the directional vector L. We may continue here,

and we'll get the sum.

Now, let us recall

the formula which gives them in the product of two vectors.

Given RN space when we multiply as an inner product of two vectors X and Y,

we use the formula.

So, these are two vectors.

We calculate the inner product by choosing

Ith coordinates of both vectors and after multiplying them,

we sum up from I one up to N. So,

that's the formula for inner product.

When we compare it with this formula,

we see that this is also an inner product.

Now, what are the vectors?

As for L vector,

we dealt with respect to earlier and this is

a directional line which passes through X zero point.

As for the components of this vector,

these are components of the vector which is widely used in multidimensional calculus.

It is also used in a lot of applications.

It is called gradient of the function F. So,

let's provide a definition.

So, this is important concept gradient

where X is a vector.

So, by definition, gradient is a vector and we

can denote it with the word grad.

This is a vector whose coordinates

are partial derivatives and they are calculated at some point.

Actually here, when we're discussing the rate of change of the function,

we are considering X zero point.

So, back to the definition this is a vector written in the form of a row.

Middle partial derivatives taken at X zero point,

this is the first derivative with respect to X one.

We continue until we reach the Nth,

and this is exactly the vector which is in the formula.

So, we can write here

a triangular bracket grad fl.

So, going back to this formula,

we see that we've got our answer.

I'm simply rewriting from there,

but I have more space here to indicate that this is

a gradient calculated at X zero point.

So, here we have grad F.

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