yousif-hag-ahmed
commented
2 months ago I apologize for that. Let's adjust the math equations to use a format compatible with GitHub's Markdown. GitHub uses backticks and single dollar signs for inline math and double dollar signs for display math. Here's the corrected content:
Definitions Glossary
- Function: A relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
- Derivative: A measure of how a function changes as its input changes. It represents an instantaneous rate of change.
- Integral: A mathematical object that represents the area under a curve. It is the opposite of differentiation.
- Partial Derivative: A derivative where we hold some variables constant and differentiate with respect to one variable.
- Gradient: A vector of partial derivatives which points in the direction of the greatest rate of increase of the function.
- Double Integral: An integral over a two-dimensional area.
- Triple Integral: An integral over a three-dimensional volume.
Differential Calculus
Definition
Differential Calculus is the study of how functions change when their inputs change. The primary object of study is the derivative.
Examples
Example 1: Basic Derivative
Find the derivative of the function ( f(x) = x^2 ).
Solution:
$$ f'(x) = \frac{d}{dx}(x^2) = 2x $$
Example 2: Trigonometric Function
Find the derivative of the function ( f(x) = \sin(x) ).
Solution:
$$ f'(x) = \frac{d}{dx}(\sin(x)) = \cos(x) $$
Example 3: Product Rule
Find the derivative of the function ( f(x) = x^2 \sin(x) ).
Solution:
Using the product rule ( (uv)' = u'v + uv' ):
$$ f(x) = x^2 \sin(x) $$
$$ f'(x) = \frac{d}{dx}(x^2)\sin(x) + x^2\frac{d}{dx}(\sin(x)) $$
$$ f'(x) = 2x \sin(x) + x^2 \cos(x) $$
Integral Calculus
Definition
Integral Calculus is the study of the area under curves. The primary object of study is the integral.
Examples
Example 1: Basic Integral
Find the integral of the function ( f(x) = x ).
Solution:
$$ \int x \, dx = \frac{x^2}{2} + C $$
Example 2: Exponential Function
Find the integral of the function ( f(x) = e^x ).
Solution:
$$ \int e^x \, dx = e^x + C $$
Example 3: Integration by Parts
Find the integral of the function ( f(x) = x e^x ).
Solution:
Using integration by parts ( \int u \, dv = uv - \int v \, du ):
Let ( u = x ) and ( dv = e^x dx ).
Then, ( du = dx ) and ( v = e^x ).
$$ \int x e^x \, dx = x e^x - \int e^x \, dx $$
$$ = x e^x - e^x + C $$
$$ = e^x (x - 1) + C $$
Multivariable Calculus
Definition
Multivariable Calculus extends the concepts of single-variable calculus to functions of multiple variables.
Examples
Example 1: Partial Derivative
Find the partial derivatives of the function ( f(x, y) = x^2y + 3xy^2 ).
Solution:
$$ \frac{\partial f}{\partial x} = 2xy + 3y^2 $$
$$ \frac{\partial f}{\partial y} = x^2 + 6xy $$
Example 2: Gradient
Find the gradient of the function ( f(x, y) = x^2 + y^2 ).
Solution:
$$ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) $$
$$ \nabla f = (2x, 2y) $$
Example 3: Double Integral
Evaluate the double integral ( \iint_R xy \, dA ), where ( R ) is the rectangle ( 0 \leq x \leq 1 ) and ( 0 \leq y \leq 2 ).
Solution:
$$ \iint_R xy \, dA = \int_0^1 \int_0^2 xy \, dy \, dx $$
First, integrate with respect to ( y ):
$$ \int_0^2 xy \, dy = x \left( \frac{y^2}{2} \right)_0^2 = x \left( \frac{4}{2} - 0 \right) = 2x $$
Now, integrate with respect to ( x ):
$$ \int_0^1 2x \, dx = 2 \left( \frac{x^2}{2} \right)_0^1 = 2 \left( \frac{1}{2} - 0 \right) = 1 $$
Make sure to review these concepts and examples thoroughly, and practice additional problems to solidify your understanding. Happy studying!
Sure, let's wrap the equations with double dollar signs for proper formatting in Markdown and add a bit more detail where needed.
Definitions Glossary
Differential Calculus
Definition
Differential Calculus is the study of how functions change when their inputs change. The primary object of study is the derivative.
Examples
Example 1: Basic Derivative
Find the derivative of the function $$ f(x) = x^2 $$.
Solution:
$$ f'(x) = \frac{d}{dx}(x^2) = 2x $$
Example 2: Trigonometric Function
Find the derivative of the function $$ f(x) = \sin(x) $$
Solution:
$$ f'(x) = \frac{d}{dx}(\sin(x)) = \cos(x) $$
Example 3: Product Rule
Find the derivative of the function $$ f(x) = x^2 \sin(x) $$.
Solution:
Using the product rule $$ (uv)' = u'v + uv' $$:
$$ f(x) = x^2 \sin(x) $$