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A partial derivative is the derivative in respect to one dimension. You can use the rules and tricks of normal differentiation with partial derivatives if you hold the other variables as constants, but the actual definition is very similar to the definition of a normal derivative. In respect to x, it looks like:

fx(x,y)=[f(x+Δx,y)-f(x,y)]/Δx

and in respect to y:

fy(x,y)=[f(x,y+Δy)-f(x,y)]/Δy

Here's an example. take the function z=3x2+2y

we want to find the partial derivative in respect to x, so we can use basic differentiation techniques if we treat y as a constant, so zx'=6x+0 because the derivative of a constant (2y in this case) is always 0. this applies to any number of dimensions. if you were finding the partial in respect to a of f(a,b,c,d,e,f,g), you would just differentiate as normal and hold b through g as constants.

Q: Definition of partial differential equation with example?

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ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.

Yes, it is.

An ordinary differential equation is an equation relating the derivatives of a function to the function and the variable being differentiated against. For example, dy/dx=y+x would be an ordinary differential equation. This is as opposed to a partial differential equation which relates the partial derivatives of a function to the partial variables such as d²u/dx²=-d²u/dt². In a linear ordinary differential equation, the various derivatives never get multiplied together, but they can get multiplied by the variable. For example, d²y/dx²+x*dy/dx=x would be a linear ordinary differential equation. A nonlinear ordinary differential equation does not have this restriction and lets you chain as many derivatives together as you want. For example, d²y/dx² * dy/dx * y = x would be a perfectly valid example

All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.

Some partial differential equations do not have analytical solutions. These can only be solved numerically.

Related questions

An ordinary differential equation (ODE) has only derivatives of one variable.

ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.

Yes, it is.

An ordinary differential equation is an equation relating the derivatives of a function to the function and the variable being differentiated against. For example, dy/dx=y+x would be an ordinary differential equation. This is as opposed to a partial differential equation which relates the partial derivatives of a function to the partial variables such as d²u/dx²=-d²u/dt². In a linear ordinary differential equation, the various derivatives never get multiplied together, but they can get multiplied by the variable. For example, d²y/dx²+x*dy/dx=x would be a linear ordinary differential equation. A nonlinear ordinary differential equation does not have this restriction and lets you chain as many derivatives together as you want. For example, d²y/dx² * dy/dx * y = x would be a perfectly valid example

All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.

All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.

Some partial differential equations do not have analytical solutions. These can only be solved numerically.

PDE stands for Partial Differential Equation

It is one in which there is only one independent variable, ie there are no partial derivatives. For example, (dy/dx) + 2y = cosx + x

The abbreviation PDE stands for partial differential equation. This is different from an ordinary differential equation in that it contains multivariable functions rather than single variables.

Poisson's equation is a partial differential equation of elliptic type. it is used in electrostatics, mechanical engineering and theoretical physics.

It's all around you, starting with equation of diffusion and ending with equation of propagation of sound and EM waves.