clairaut said:I can't see the proof clearly.
Gradient [f(tx,ty)] dot
<d/dt (tx) , d/dt (ty)> =
d/dt [f(tx,ty)] = n * t^(n-1) * [f(x,y)]
I don't see how the t^(n-1) term disappears.
HELP
A homogeneous function of degree n is a mathematical function in which each term has the same degree n when all variables are raised to the same power. This means that if you multiply all the variables by a constant, the resulting function will also be multiplied by that same constant.
Homogeneous functions are important in mathematics because they exhibit certain properties that make them useful in solving equations and studying relationships between variables. They also have applications in areas such as economics, physics, and engineering.
To determine if a function is homogeneous of degree n, you can use the Euler's homogeneous function theorem. This theorem states that if a function f(x1, x2,...,xn) is homogeneous of degree n, then it satisfies the equation x1∂f/∂x1 + x2∂f/∂x2 + ... + xn∂f/∂xn = nf, where ∂f/∂xi represents the partial derivative of f with respect to xi.
No, a function can only be homogeneous of one degree. This is because the degree of a homogeneous function is defined as the sum of the exponents on all the variables, and if the function had multiple degrees, it would not satisfy the property of homogeneity.
Homogeneous functions and homogeneous polynomials are closely related. A homogeneous polynomial is a polynomial in which all terms have the same degree, and thus can be considered as a special case of a homogeneous function. However, not all homogeneous functions are polynomials, as they can also involve other mathematical operations such as logarithms and trigonometric functions.