These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.
The Life and Work of Sofya Kovalevsky
“Say what you know, do what you must, come what may.”
Sofya Kovalevsky 
As female college students in the 21st century, there are often things we take for granted. One of the most obvious of these is the simple right to go to school and study what we choose. When we study the lives of women who have come before us, our freedom seems to be more of a treasure. One woman who fought the social standards of her society in order to study the subject she desired was Sofia Kovalevskaya.
Sofia Krukovsky Kovalevskaya, often called Sonya by close friends and family, was born into an aristocratic family in Moscow in 1850. Her parents, General Vasiliy Krukovsky and his wife Elizaveta, gave birth to their second child Sonya in their seventh year of marriage. Sonya, the youngest girl in the family, had an older sister, Anyuta, who was later to become one of her best friends and her role model, and a younger brother, Fedya .
When the General’s military career ended in 1858, he moved the family to an estate near the Lithuanian border called Palibino. It is here that Sonya would begin regular schooling and eventually develop a love for mathematics. It is said that the Krukovskys ran out of wallpaper when decorating their new home and resorted to General Krukovsky’s old Calculus notes for the walls in the nursery. Later Sonya claimed that the paper allowed her to become more familiar with the calculus symbols even before learning the material .
Although Sonya grew up with a childhood tutor named Joseph Malevich who taught her basic arithmetic, she credited most of her interest in sciences and mathematics to her two uncles. Fyodor, her mother’s younger brother, would spend time with Sonya and tell her stories about biology. Vasily’s brother Pyotr was not a teacher or even well educated, but he and Sonya would talk together about mathematical concepts, such as squaring circles and asymptotes. It was these conversations that would kindle Sonya’s interest in the mathematical sciences .
As Sonya’s academic interests increased, so did her father’s disapproval. At age thirteen, in order to continue studying mathematics, Sonya smuggled an Algebra text into her room to study. After mastering the material, Sonya wanted to study Physics. Her neighbor, Professor Tyrtov, decided to permit her to study his book if she would teach herself trigonometry, which was needed for the section on optics. When Tyrtov finally observed her work, he was amazed and called her “a new Pascal.” Sonya had explained the sine function by constructing a chord on a circle using a development strikingly similar to that of the actual development of the mathematics. Tyrtov was so impressed with the intelligence of the girl that he convinced General Krukovsky to allow Sonya to go to St. Petersburg to study mathematics under Professor Alexander Strannoliusky .
After finishing her studies around 1867, Sonya became determined to further pursue her education. Since the universities in Russia were closed to women, Sonya’s only option was to travel west.
This task, however, seemed to be insurmountable. Sonya Krukovsky was living in a society where women were completely dependent upon men. By law, a woman traveling alone had to have either her father or husband’s permission. General Krukovsky seemed unyielding. He would not allow his daughters to leave the country to study. The only option left available to Sonya was to marry a husband who would allow her to go.
During the time when Sonya had become overwhelmed with her studies, Anyuta, her sister, had become an advocate for the liberation of women. In her discussions with some of the other revolutionaries of her generation, Anyuta found a solution to the problem of not being able to study abroad, a “fictitious husband” by the name of Vladimir Kovalevsky .
Vladimir, a twenty-six year old university student studying geology, became mesmerized with Sofia, and in September of 1868, they were married. Soon afterward, they left for Petersburg to study, and to perhaps find a suitable husband for Anyuta. While at Petersburg, Sofya received unofficial permission to attend classes, but this did not ensure her equal treatment. In a letter to her sister she wrote, “…Friends will solemnly escort me by way of the backstairs so that there is hope of hiding from the administration and from curious stares.” Sofya was determined, and it was at Petersburg that she decided to make mathematics her life’s ambition .
In 1869, the Kovalevsky’s and Anyuta left Petersburg and traveled to Heidelberg, Switzerland. Here, Sofya planned to study mathematics, but once again was held back. At first, the university would not allow Sonya to matriculate, but soon after an appeal, she was given permission to attend classes at the discretion of each professor .
After studying for a year at the university in Heidelberg, Sonya’s thirst for mathematical knowledge remained unquenched. She decided to seek out one of “the most noted mathematicians of the time,” Karl Weierstrass . This quest led her to the University of Berlin, where once again she discovered that female students were not welcome. Even after Weierstrass saw her work and personally requested that she be allowed to attend his classes, the university denied her this opportunity .
Because Weierstrass recognized Kovalevsky’s great mathematical ability, he offered to tutor her privately. This working relationship lasted for four years, and the resulting friendship continued throughout most of her life. While under the tutelage of Weierstrass, Sonya completed three papers for a doctoral dissertation. They were: “On the Theory of Partial Differential Equations,” “On the Reduction of a Certain Class of Abelian Integrals of the Third Rank to Elliptic Integrals,” and “Supplementary Remarks and Observations on the Laplace’s Research on the Form of Saturn’s Rings.” Finally, in July 1874, Sonya Kovalevksy’s hard work was rewarded, and she received a Ph.D. in absentia, suma cum laude, from the University of Gottingen. She became the first woman to receive her doctorate in mathematics .
Even after earning her Ph.D., Kovalevsky was unable to obtain a job. As a result, Sonya returned home to Russia with Vladimir, Anyuta, and Anyuta’s husband. For the next five years, Sonya lived the life of a typical Russian woman. She and Vladimir finally began to live as husband and wife, and in October of 1878, Sonya gave birth to a daughter, Sofya Vladimirovna .
Sonya found family life to be unfulfilling. She longed for freedom to study, and in 1880 her wish was granted. Sonya Kovalevsky was invited by Russian mathematician Chebyshev to present a paper at a scientific conference in Sweden. Although she had not done research in several years, Sonya was able to use her paper on Abelian Integrals, one she had written while being tutored by Weierstrass. Her presentation was such a success that she was invited to teach at a university in Sweden .
Before Kovalevsky moved to Sweden, she secretly visited Weierstrass in Berlin and once again began to actively research mathematics. As a result of this decision, Sonya’s marriage ended. Faced with the dilemma of being a single mother, Sonya chose to leave her daughter in the care of friends and took refuge in her work .
In 1883, Sonya took on her first job as a lecturer at the university in Stockholm, but this did not mean that she was treated as an equal to her male counterparts. Sonya was forced to begin her job on probation and without pay, but she proved herself mathematically and was eventually appointed as a Professor of Higher Analysis. In addition to this, she became an editor for Acta Mathematica and published a paper on crystals .
After Sonya was sure of her position at the University of Stockholm, she brought her daughter to live with her in Sweden. In the same year she was appointed to the Chair of Mechanics and published a second paper on the propagation of light in crystals. But this paper proved to contain a serious flaw – Sonya had incorrectly used a function in her calculations. Sonya argued that Weierstrass should have discovered the error, but her tutor claimed to be ill and fatigued .
In the fall of 1887, Sonya’s sister Anyuta died. At the news of this, Sonya was greatly distressed and sought once again to find solace in her work. The French Academy of Science announced the Prix Bordin in 1888, a competition in which contestants were to present their papers on the theory of the rotation of a solid body. Sonya became intrigued by the topic and decided to enter the contest .
As her scientific life began to evolve, Sonya’s personal life did as well. Maxim Kovalevsky, a Russian lawyer arrived in Stockholm. Maxim was to become not only one of Sonya’s close friends, but also her lover.
Although Sonya struggled between researching and spending time with Maxim, she succeeded in completing her paper for the Prix Bordin on time. To fairly determine a winner, the fifteen papers were submitted anonymously. Authors wrote a quote on their paper to be used for later identification . One of the papers proved to be so impressive that the reward was increased from 3,000 to 5,000 francs . Surprisingly, the winner was a woman, Sonya Kovalevsky. Her work on the motion of an unsymmetrical rigid body about a fixed point was the first to have been successfully completed.
Kovalevsky’s paper was a continuation of the work of Leonard Euler and Joseph-Louis Lagrange. In the late 1700s, Euler focused on a model where the center of mass is at the fixed point. This can be demonstrated by a spinning top, where the center of mass is at the point where the top spins. Lagrange came up with another model in 1788, in which he modified Euler’s work by exploring a top with the center of mass at the center of the top. This model is shaped similarly to the tops we spin in everyday life .
In each of these instances, the mathematicians were trying to discover equations to
accurately describe the rotating motion of the models. In order to do this, it was
imperative that they use partial differential equations. A partial differential equation is an
equation that includes derivatives of a function with respect to more than one variable.
One example of this type of equation is δ2u + δ2u = 0. The solution to this
equation is simply a function whose partial derivatives satisfy this equation.
A simpler type of equation using derivatives is known as a differential equation.
Instead of including the partial derivatives of a function, a differential equation shows the
relationship between a function and its derivative. The solution to one of these
equations is the function that satisfies all values of the equation. An example of a
differential equation is dy = y + 1 . One solution to this equation is y(t) = 2t + 1. This
dt t + 1
can be shown by taking the derivative of y(t), which is 2. When substituting y(t)
into the original equation, you get, dy = (2t + 1) + 1 . After simplification, dy/dt = 2.
dt t + 1
Since y’(t) = dy/dt, y(t) satisfies the differential equation and is a solution.
For many years, no partial differential equation could be found to solve the problem of an unsymmetrical rotating solid. For her paper in 1888, Sonya found a solution. In her case, Kovalevsky used a solid similar to Euler’s but added a heavy object to one side of the model. She then created new equations and a different and much more difficult integral. Kovalevsky had taken an old problem and used a new approach to solve it .
The following pictures depict the Euler, Lagrange-Poisson, and Kovalevsky tops respectively .
Despite all her accolades, Sonya was unable to obtain a job in Russia. Even after being given honorary membership in the Russian Academy of Sciences, her search was fruitless. Sonya’s only option became the renewal of her professorship at Stockholm .
While working in Stockholm, Sonya often traveled to visit Maxim in France. During these visits, he encouraged her to write about her life. The result of this was two novels: Memories of Childhood and The Nihilist Girl. Sonya was never able to complete her second novel. She contracted pneumonia from her frequent travel, and died at the age of forty-one .
Sonya Kovalevsky left a mathematical legacy. She is attributed with ten papers in mathematics and mathematical sciences, and has left a lasting impact on the standing of women in academic society. Topics she explored included: the propagation of light through crystals, Abelian Integrals, Differential Equations, and Saturn’s Rings . Sonya is most known for her work on the rotation of a rigid body. Some have described Sonya Kovalevsky as, “the brightest star among female mathematicians since the time of Hypatia” .
1. A quotation by Sofia Kovalevskaya web page,
Comments: The quote given at this address is the one that Kovalevsky used on her paper for the Prix Bordin.
2. Cooke, Roger. “S.V. Kovalevskaya’s Mathematical Legacy: The Rotation of a Rigid
Body.” Vita Mathematica, The Mathematical Association of America.
Comments: This article was somewhat useful in understanding the background of the work Kovalevsky did on the rotation of a rigid body.
3. Karl Wilhelm Theodor Weierstrass web page,
Comments: This web page is useful in understanding more about who Weierstrass was and why it was important that he was Sonya’s tutor.
4. Kovalevsky, Sofia. “A Russian Childhood.” Springer-Verlag, 1978.
Comments: This article contains good explanations and excellent diagrams of the work done by Euler, Lagrange, and Kovalevsky.
5. Rappaport, Karen D. “S. Kovalevsky: A Mathematical Lesson.” The American
Mathematical Monthly 88 (October 1981): 564-573.
Comments: This article was extremely useful in understanding Kovalevsky’s life and her struggles as a woman mathematician.
6. Sofia Kovalavskaya web page, by Becky Wilson, Agness Scott College,
Comments: This article was not used in the paper, but may be helpful in understanding the Kovalevksy’s life.
7. Sonya Kovalevski web page,
Comments: A quotation from this article was used in the paper.
8. Tabor, Michael. “Modern dynamics and classical analysis.” Nature, Vol 310 (July
Comments: This article is useful for understanding both Kovalevksy’s life and work, but was not used in this paper.
9. The Works of Sonya Kovalevskaya web page, by Kimberly A. Meares,
Comments: This article was useful in understanding the way the Prix Bordin competition worked.
10. Who was Sonia Kovalevsky web page, adapted from David M. Burton’s History of
Comments: This article discusses the accomplishments of Sonya Kovalevsky’s life. It was useful in determining that Kovalevsky was the first woman to receive a Ph.D. in mathematics.