Gradient Vector and Linearization

Gradient Vector

The gradient vector is a vector of all partial derivatives of a function. It is defined by the differential operator ${\nabla = ( \frac{\partial}{\partial x}, \frac{\partial}{\partial y},...)}$ . The gradient vector of a function ${f}$ is ${\nabla f= (f_x, f_y,...)}$ .

For a surface given explicitly by ${z=f(x,y)}$ , ${\nabla f}$ is a 2 dimensional vector pointing the direction of steepest increase.

For a surface given implicitly by ${f(x,y,z)=0}$ , ${\nabla f}$ is a 3 dimensional vector that is normal to the surface. We can then compute a tangent plane using point normal form ${(\mathbf{r}-\mathbf{r_0})\cdot \mathbf{n} = 0}$ where ${\mathbf{n}=\nabla f(\mathbf{r_0})}$ (see previous page for example).

Linearisation and Differential Form

In single variable calculus we use a tangent line to linearly approximate a function and compute estimates of function values. In multivariable calculus we use a tangent plane to obtain the linearised change in a function ${f(x,y)}$ . To have a tangent plane at a given point the function must be differentiable at that point.

A function ${f(x,y)}$ is differentiable at ${(x_0, y_0)}$ if ${f_x}$ and ${f_y}$ exist near ${(x_0, y_0)}$ and are continuous at ${(x_0, y_0)}$ .

Thus for the function ( ${\Delta}$ means change):

$\displaystyle \Delta f(x,y)=f(x,y)-f(x_0,y_0)=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y +\epsilon_1 \Delta x+\epsilon_2 \Delta y$

Where ${f_x...+f_y...}$ is the linearised change in ${f}$ and ${\epsilon_i}$ is the error which ${\rightarrow 0}$ as ${(x,y)\rightarrow (x_0,y_0)}$ .

In the limit as ${\Delta x, \Delta y \rightarrow 0}$ the total differential is ${df = \frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy}$ , where ${df,dx,dz}$ are infinitesimal differentials.

Note: Also see later section on Taylor series for more on approximations.

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