Multivariable Calculus

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What is Multivariable Calculus

In Mathematics, multivariable calculus is also known as multivariate calculus. Multivariable calculus is the study of calculus in one variable to functions of multiple variables. The differentiation and integration of multivariable calculus include two or more variables, rather than a single variable.

Multivariable calculus is a branch of mathematics that helps us to explain the relation between input and output variables. For example, if the output of your function z is dependent on one input variable i.e. z, then it gives us

Z = f (x)

Similarly, if the output of your function z is dependent on more than one input variable i.e. x, and y then it gives the function as 

Z = f( x, y)

The variables x and y are the input of function, hence they can influence the result of output.


How to Solve Multivariable Calculus?

  • If a function is dependent on multiple variables, then we can use partial derivatives,  to determine the derivative of a function concerning to one of those variables. The trick that we have to follow here is to keep all the variables constant.

  • If we change all the variables and find the derivative, then it will be considered as a total derivative.

If there are two function f(x), and g(x), and let us also consider that the derivative of both the functions can be calculated, then the product of their derivative will be

(f + g)' = f' + g'

Hence, the derivative will be the sum of the derivative of a function f and g.


Basic Multivariable Calculus

Basic multivariable calculus is the study of integration and differentiation of two or more variables. Basic multivariable calculus introduces two types of calculus known as integral calculus and multivariable calculus. Both of these concepts are based on the idea of limit and continuity. Differential calculus helps us to find the rate of change of quantity whereas integral calculus helps us to determine the quantity when the rate of change is known.


Advance Multivariable Calculus

Advance multivariable calculus is just a fancy method of briefing the topic in calculus that requires a bit more thought and work. Advance multivariable calculus is just the application of some basic multivariable principles like differentiation, integration, rate of change, etc. Vector space, linear transformation, and matrices are some important areas of multivariable calculus.


Some of the Topics Included in Advance Multivariable Calculus Are

  • Quadratic Form

  • Integration Form

  • Vector Field

  • Curves and Surface

  • Critical point analysis for multivariate function

  • Gradient's theorem for the line integral, Green's theorem, Stokes' theorem, and the divergence theorem.


Multivariable Differential Calculus

Multivariable differential calculus is similar to the differentiation of a single variable. As we move up to consider more than one variable, things work quite similarly to a single variable, but some small differences can be seen.

Given the function z = f ( x, y), the differential dz or df is derived as

dz= fₓdx + fydy or df = fₓdx + fydy

There is a natural expansion to the function of three or more variables. For example, given the function w = g( x, y, z), the differential is given by 

 dw= gxdx + gydy + gzdz


Multivariable Differential Calculus Example

Find the differential of Z = p³q⁶/r²

Solution:

dz= 3 p² q⁶/r² dp + 6 p³ q⁵ / r² dq- 2p³q⁶/r³ dr


Solved Example

  1. Find the first partial derivative of function z = f(p,q) = p³ + q⁴ + sin pq, using curly dee notation.

Solution:

Given Function: z = f( p,q) = p³ + q⁴ + sin pq

For a given function, the partial derivative with respect of p is

∂z/∂p = ∂f/∂p = 3p² + cos( pq) q

Similarly, the first he partial derivative with respect of q is 

∂z/∂q = ∂f /∂q = 4q³ + cos( pq) p


  1. Find the total differentiation of the function : Z = 2p sin q - 3p²q²

Solution: 

Given 

Function: Z = 2p sin q - 3p²q²

The total differentiation of the above function is derived as

dz= ∂z/∂p dp+ ∂z/∂q dq

dz = ( 2 sin p - 6pq²)dp

= +(2pcosq - 6p²q)dq


FAQ (Frequently Asked Questions)

1. What is calculus?

In Mathematics, calculus is the study of the rate of change. Generally, calculus is used in Mathematical models to obtain feasible solutions. It also helps us to understand the variation between the values which are related by function. The two types of calculus are:

  • Differential calculus

  • Integral calculus

2. What are the applications of multivariable calculus?

Multivariable calculus is applied in different fields such as Economics, Engineering, Physics, Computer Science, etc. Some of different applications of multivariable calculus are as follows:

  • Multivariable calculus is applied in different fields such as Economics, Engineering, Physics, Computer Science, etc. Some of the different applications of multivariable calculus are as follows:

  • Multivariable calculus is used in optimal control of a continuous-time dynamic system.

  • Quantitative data in finance also uses multivariable calculus to estimate future trends in the stock market.

  • Multivariable calculus in Engineering and Social Science is used to study and model the high dimensional structure that manifests the deterministic nature.

3. What is the difference between the single and multi variables calculus?

The difference between single and multivariable calculus is that single calculus deals with just only one function whereas multivariable calculus deals with the function of more than two variables. Differential and Integration of both single and multivariable calculus is quite similar. In multivariable calculus, If a function is dependent on multiple variables then we can use partial derivatives to determine the derivative of the function with reference to one of those variables while holding everything constant.


A more interesting part of multivariable calculus is vector calculus which consists of topics like Green's theorem, Stroke theorem, Gradients, Divergence theorem, etc.

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