Mathematical tools for Environmental EngineeringLaajuus (5 ECTS)

Objective

After the course student recognizes the most used mathematical tools for environmental engineer.
The student understands the meaning of the integral of a function graphically and algebraically on own fields applications.
The student is able to use differential and intregral calculus in his or hers subsequent studies in physics, chemistry, biology and professional studies, and in subsequent courses in mathematics.
The student is able to interpret vectors expressing either state, position, change, or direction.
The student is able to carry out elementary algebraic operation of vectors. In addition he or she can visualize those operations in two dimensional space.
The student is able to apply 2D ja 3D vectors in chemical, environmental or physical problems.
The student recognizes, and can list typical examples of higher dimensional vectors in analytical chemistry and in environmental engineering.
The student is able express dependencies using tools of analytical geometry and linear (empirical) models.
The student can explain what is meant by a function and can apply functions in basic engineering problems.
The student can graph typical functions appearing in engineering problems.

Content

1. The concept of a limit of a function, and limits of typical functions appearing in science and in environmental engineering.
2. Derivative, numerical differentiation the use of differentiation rules.
3. Newton’s method for equations.
4. The integral function and the concept of an integral and its relation to the integral function.
5. Numerical integration methods.
6. Applications of derivatives and integrals in science and environmental engineering.
7. The concept of a gradient.
8. Using mathematical software in differential and integral calculus.
1. Vector algebra
2. Scalar, vector and triple products, and projections
3. Straight lines, planes and hyper-planes
4. Linear and quadratic empirical models
5. Functions
6. Computer aided mathematics

Assessment criteria, satisfactory (1)

1. The student is able to perform simple algebraic operations on vectors, such as addition, subtraction, multiplying by a scalar, and calculating norms. The student can also visualize the aforementioned operations in 2D.
2. The student is able to apply scalar and vector products in calculating angles between vectors, and calculating areas and volumes of simple geometrical objects. The student is able to calculate scalar and vector projection in any dimension.
3. The student is able calculate equidistant points along a given straight line, and the student is able to define the equation of a plane passing given 3 points.
4. The student can graph typical basic functions appearing in environmental engineering.
5. The student can find out limits of elementary functions.
6. The student can explain the derivative of a function in geometrical terms
7. The student can explain the integral of a function in geometrical terms.
8. The student recognizes possibilities of derivatives and integrals in simple problems of chemistry, physics and biology.

Assessment criteria, good (3)

1. The student can apply vectors in physical 2D or 3D problems of movement or force balances.
2. The student can apply scalar and vector products in calculating properties of real geometrical objects.
3. The student is able apply straight lines either in environmental processes.
4. The student can apply the concept of a plane in linear empirical modeling.
5. The student can determine inverse and composite functions.
6. The student shows that he or she is apply mathematical software in visualizing vectors or geometrical objects.
1. The student can find out limits of rational functions.
2. The student can explain the derivative of a function in algebraic terms and can approximate derivatives using forward or symmetric difference quotient.
3. The student can use Newton’s method for solving simple equations that don’t have analytical solutions.
4. The student can explain the integral of a function as a limit. The student can determine integral functions of typical functions appearing in environmental engineering.
5. The student can apply several numerical integration techniques.
6. The student can use derivatives and integrals in simple engineering or science problems, e.g. in optimization.
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8. The student shows ability to use mathematical software in calculus problems.

Assessment criteria, excellent (5)

1. The student can explain the difference between positional (state) and change (directional) vector in a given practical application.
3. The student is able apply straight lines defining new states of a process both distance based or variable change based.
4. The student is able to interpret linear empirical models appearing in environmental research articles.
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2. The student knows techniques for differentiating data that are given in tabular form, e.g. spectra.
3. The student can use Newton’s method for solving equations that don’t have analytical solutions
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5. The student can integrate data that are given in tabular form, e.g. spectra.
6. The student can use derivatives and integrals engineering problems, e.g. in in problems of related rates.
7. The can explain how gradients can be used in optimization of functions of several variables.
8. The student shows ability write programs in Matlab or in some other Mathematical software.

Assessment criteria, approved/failed

1. The student is able to perform simple algebraic operations on vectors, such as addition, subtraction, multiplying by a scalar, and calculating norms. The student can also visualize the aforementioned operations in 2D.
2. The student is able to apply scalar and vector products in calculating angles between vectors, and calculating areas and volumes of simple geometrical objects. The student is able to calculate scalar and vector projection in any dimension.
3. The student is able calculate equidistant points along a given straight line, and the student is able to define the equation of a plane passing given 3 points.
5. The student can graph typical basic functions appearing in environmental engineering.
1. The student can find out limits of elementary functions.
2. The student can explain the derivative of a function in geometrical terms and can differentiate elementary function using the differentiations rules.
3. --
4. The student can explain the integral of a function in geometrical terms. The student can determine integral functions of elementary functions.
5. The student can apply the center-point numerical integration technique.
6. The student can use derivatives and integrals in simple problems of chemistry, physics and biology.